philosophy of physics

What is “Classical Mechanics” Anyway? In this formula, mi is the mass of the partic le numbered as i, qi(t) is its vec tor loc ation at time t and the various fi (j, t) will supply the strengths of spec ific forc es applic able to partic le i that have their origins in a partic le j (for i ≠ j). But this merely lays down the basic scaffolding we will need for a properly c ompleted equational set. In partic ular, we must yet spec ify the sundry fi(j,t) in c onc rete ways that c an lead to a set of equations that are uniquely solvable (at least most of the time) with respec t to an arbitrary set of initial c onditions (supplementary “data” that provides qi(t0 ) and dqi(t)/dt| t0 information for eac h partic le i ∈ S and some partic ular time t0 ).To ac hieve suc h formal c losure within a c ompleted rec ipe, eac h fi(j,t) must represent a special force law that links the strength of the fi(j,t) forc es somehow to the q- loc ations of partic les i and j (as we will see, Newton's third law puts sharp restric tions on the nature of this dependenc y). The basic prototype for a “spec ial forc e law” of this ilk is Newton's law of gravitation: but allied princ iples are needed to govern all other applic able forc es (suc h as those responsible for the c ohesion and repulsion of matter). Eac h additional law is expec ted to c arry in its wake its own range of material constants, suc h as the charges c (i) that show up in the static form of Coulomb's law19 : (where suc h c (i) c an sometimes c arry negative values).We are said to have supplied a constitutivemodeling2 0 for the system S in point-mass terms onc e we have spec ified S's full c omplement of partic les i and the values of the applic able “forc e law” c onstants m(i), c (i), and so on (the list of c onstants then tells us how many fi(j,t) terms are “turned on” within S). One of the most frustrating aspec ts of the c lassic al point-mass tradition is that it never fully resolved what these spec ial “forc e laws” (besides gravitation) should be. Modern molec ular modelers frequently utiliz e sundry mixtures of sixth and twelfth power princ iples (e.g., the familiar Lennard-Jones potential) between point masses to simulate the molec ular interac tions within a gas, but no one maintains that suc h rough rules enjoy any c anonic al status within c lassic al mec hanic s. This inc ompleteness trac es to the fac t that nature has indic ated no spec ial preferenc e for classical princ iples governing, say, small-sc ale c ohesion and repulsion bec ause it has dec ided to let matter behave in a strongly quantum mec hanic al manner in suc h c lose quarters. So molec ular modelers are left with a rather diffuse c ollec tion of princ iples that might possibly model c lose range interac tion ably, with the final c hoic e being dec ided by what appears to work. Indeed, textbooks frequently sidestep the need to fill in the spec ial-forc e- law holes in an Euler's rec ipe modeling through various forms of evasion, suc h as the appeals to rigid-body c onstraints. Typic ally, suc h diversionary appeals tac itly shift us into ontologic al realms natural to c ontinua and rigid bodies, whic h approac h the problems of c ohesion in an inherently different manner than c urrently c ontemplated. The absenc e of enough spec ial forc e laws in the point-mass setting engenders another familiar diffic ulty for point- mass mec hanic s that is usually “solved” by shifting the underlying ontologic al framework. Suppose we c onstruc t a constitutive modeling for the solar system, where we treat the sun and planets as point masses and the only spec ial forc e we turn on is gravitation. The resulting Euler's-rec ipe equational set will be “formally well-posed” in the sense that we are supplied the right number of equations to potentially possess unique solutions given initial conditions. But that is merely a “formal” guarantee in the sense that it tells us that we are somewhere in the ballpark of getting the solutions we wanted. It does not c ompletely assure us that the solutions really exist. And we have good grounds for worrying about this. Roc ket designers apprec iate the fac t that one c an supply a projec tile with a signific ant inc rease in kinetic energy by slingshoting it through the strong gravitational field of a planet (the tec hnique was used several times to generate enough boost to propel the Cassini spac e probe to Saturn) (figure 2.9). Page 12 of 45

What is “Classical Mechanics” Anyway? Figure 2.9 Sinc e point masses have no siz e, and bec ause we have not inc luded any sort of repulsive forc e in our equational set, c an a partic le possibly2 1extrac t enough energy from its planetary near approac hes to produc e an infinite velocity boost within a finite span of time? This was a famous mathematical question that was settled in the affir- mative by Zhihong Xia in the 1980s. As such, the velocity blowup indicates that the viability of our point-mass modeling has self-destruc ted of its own ac c ord. Rather than searc hing for repulsive spec ial forc e laws that might inhibit the effec t, physic ists typic ally brush the problem aside, “Oh, you've just neglec ted the finite siz e of real planets.” That is true, but they have thereby esc aped to the dominions of rigid body or c ontinuum mec hanic s in making suc h appeals. The full battery of spec ial forc e laws that point-mass mec hanic s requires is further skirted by the c ommon prac tic e of presuming that, however the true missing laws prec isely operate, their net effec ts c an be lineariz ed or otherwise simply approximated as long as their activities are not strong (the telltale symptom of this ploy are terms in equa- tions such as a linear “Wq(x,t)” whose special-force-law origins are left hazy). Many of the descriptive successes of nineteenth-century physics were prosecuted under the guidance of such approximatizing assumptions. It is to be expected that such modelings will frequently self-generate holes in their descriptive coverage due to the fact that their solution sets can evolve into situations where the presumptive ansatz that “the activities of the unspecified force law can be approximated by Wq(x,t)” must plainly fail. It is striking that if one inspects the stock point-mass modelings provided in popular textbooks, very few of them completely satisfy the provisos of Euler's modeling recipe and instead invoke some tactic for special-force-law avoidance. Such methodologies of avoidance can be prudent in practice,i through preventing the merits of a modeling from being held hostage to the delicate specifics of an unproven special force law. Absent any definitive resolution of what its full complement of special force laws f (j,t) should comprise, our Eulerian recipe employs “F = ma” as a skeletal frame upon which a formally closed differential equation modeling might be eventually assembled as soon as adequate skin and clothes can be foun2d2 for the task. Certainly, one cannot coherently discuss issues such as whether classical mechanics is deterministic until these issues of special force laws have gotten fleshed out in some fuller manner. Special force laws represent the natural point-mass analog to the constitutive laws of modern continuum me- c hanic s. In both c ases, we must learn to watc h out for physic s avoidanc e stratagems that bypass some of the expec ted ingredients in the relevant rec ipe. As we have noted, suc h c onstitutive-modeling evasions frequently take the form of mixed sc ale-level lifts in whic h the desc riptive voc abulary natural to a higher sc ale ΔL* bec omes invoked in a manner that allows the modelers to evade the nontrivial constitutive modeling concerns they would otherwise need to confront had they remained resolutely at the original modeling scale ΔL. There are several widely disc ussed aspec ts of Newton's laws of motion that merit quic k remark. Regarding his sec ond law, implic it within our Eulerian rec ipe is the assi umption that sound modeling equations c an be set up for every system S based upon their Cartesian locations q (t) within “absolute space” or, more minimally, with respect to some c hoic e of inertial frame. Newton himself, insofar as I c an tell, never quite made suc h a c laim, for he often set up his equations using what are often c alled “natural c oordinates”—quantities that possess a palpable physic al signific anc e within the target system itself. For instanc e, in the c ase of a bead sliding along a c urved wire, the arc length along the wire qualifies as a natural coordinate, whereas the bead's location within an externally defined frame does not. Within c elestial mec hanic s, this distinc tion enjoys little purc hase, but the issue bec omes pertinent when material c onstraints suc h as “moving along a rigid wire” c ome into play. Page 13 of 45

What is “Classical Mechanics” Anyway? Regarding Newton's third law, its original formulation (as the so-c alled princ iple of “ac tion = reac tion”) seems patently haz y and has been historic ally subjec t to substantially divergent interpretations. In a modern point-mass reading, it is usually regarded as plac ing various strong restric tions upon the spec ial forc e laws we are allowed to employ in Euler's rec ipe: (3a) All forc es arise between pairs of partic les and have their sourc e in one of the pair. (3b) These forc es are direc ted along the line between the masses (the forc es are “c entral”) and opposite in magnitude (“balanc ed”). (3c) The strength of these forc es depends only upon the spatial separation between the bodies and not, say, upon their relative veloc ities. Figure 2.10 In other words, if a spec ial forc e law c laims that mass j exerts a spec ific forc e fi (j,t) upon mass i, then j must exert a rec iproc ating forc e fj (i,t) upon i equal in magnitude to fi(j,t) but reversed in direc tion (observe that only ac tion- at-a-distanc e forc es are relevant within a point-mass setting, so fi(j,t) ac ts at i's position, whereas fj (i, t) ac ts at j's position). Although Newton's own law of gravitation suits these requirements, it is unc lear that he would have ac c epted the (3a-c ) supplements in the strength stated. Requirement (3c ), for example, stands in apparent c onflic t with most varieties of fric tional forc e bec ause their strength generally depends upon the rate (dq(x)/dt) whereby bodies slip past one another. Requirement (3b) seems to rule out sheering forc es, suc h as arise when one layer of water slips over another, or the sideways forc e that a c harged partic le feels near a magnetic pole (note, however, that some of these situations only pertain to extended objec ts in c ontac t along an interfac e and may not direc tly c onc ern us now). One of the c hief reasons for making suc h strong restric tions on forc es is that they are required to establish vital tenets like Galilean relativity, balanc e of angular momentum, and the c onservation of energy within a point-mass frame (Newton did not maintain energy c onservation himself). This is bec ause the underlying notion of potential energy requires some restric tion akin to (3c ). Partially due to its vaguely expressed c ontours, Newton's third law often serves as a signific ant site of substantial lifts within mec hanic s. Let's look at a typic al example in the c ontext of a familiar sc ientific toy: a line of steel ball pendulums lying adjac ent to one another (figure 2.10). Page 14 of 45

What is “Classical Mechanics” Anyway? Figure 2.11 If the ensemble is struc k by a falling ball bR to the right, it will c ome to rest and the ball bL at the left end will fly off. But as soon as gravity pulls bL into c ollision with the group, bL will halt and bR will fly off again, to return, more or less, to its original state. And bac k and forth the knoc king osc illations will go, until fric tion eventually brings the ensemble to rest. And it is natural to c onc eptualiz e this situation in this manner. The originally falling bR externally exerts an impac tive forc e upon its first member of the adjac ent ensemble bR−1, whic h then imparts a c ongruent internal forc e upon its nearest neighbor bR−2 and so on ac ross the array until we reac h bL. Sinc e bL lac ks any leftward neighbor upon whic h to exert a leftward forc e, it is forc ed to c onvert that potential into its own kinetic motion, whic h will be of the same magnitude as bR originally possessed, under the presumption that the masses mL and mR are identic al. Expressed in ersatz third-law-style jargon, we c an say: ball bR originally supplies an impressed external forc e upon its neighbors, whic h then exc ites a spec trum of internal forc es in direc t c ontac t with another. Bec ause of the third law, these internal forc es will exac tly c anc el eac h other out in terms of any work they c an perform on the ensemble, henc e the c entral pac ket of balls will display no visible movement. But ball bL lac ks a balanc ing left-hand neighbor, so it is forc ed to c onvert the impressed forc e upon it into its own kinetic movement. Often related reasoning is presented in a somewhat more elaborate guise. Rather than allowing bR to fall against the group, let us simply push against the entire group at ball bR with an applied forc e of magnitude F (figure 2.11). What c ountervailing forc e should we apply to bL to maintain the whole group in equilibrium? −F, obviously. Let us now c onc eptualiz e bL's so-c alled “inertial reac tion” mL2 x/dt2 as a kind of “forc e” (until rec ent times, it was quite c ommon to employ the term “forc e” in this wider manner). Returning to our original “bR supplying an external forc e to the group” c ase, we c an c odify our predic tion in the guise: bL will develop an inertial reac tion forc e exac tly equal in magnitude and direc tion to −F. In this format, the reasoning of our previous paragraph c an be extended: mec hanic al systems always maintain a kind of equilibrium, wherein c ertain members will c ounter any unbalanc ed forc es upon them by forming the requisite inertial reac tions. All of this reasoning is well and good in a c ertain sense, exc ept that (1) its notion of “forc e balanc e” has nothing to do with Newton's third law as we have interpreted it and (2) reasoning of this type properly requires the realm of rigid bodies for its firm support and c an only be regarded as a rough approximative lift within the stric t c ontext of point-mass mec hanic s. To see what has gone wrong, let us replac e our array of pendulum balls with a lattic e line of legitimate point masses. To rec onstruc t a point-mass substitute for the pendulum-like behavior of the balls, we need (1) some spec ial forc e law Frep(xi , xj ) to generate repulsive forc e that point i will exert upon point j under c lose approac h and (2) some outside sourc e of attrac tive forc e Fatt(xi ) to hold eac h point mass i within a neighborhood of its lattic e rest position. Now apply a forc e F to the lattic e point pR. What does our third law, as heretofore interpreted, demand? Only that Frep(xi , xj ) =−Frep(xj , xi ) and that the unspec ified sourc es of F and Fatt(xi ) should feel rec iproc al forc es upon themselves. There is absolutely no requirement that the summed forc es upon our sundry lattic e points i will “perform no work” upon them. In fac t, this will generally be false: the initial blow will send waves of c ompression and expansion through the lattic e, at eac h stage of whic h small amounts of work will be exerted on eac h i. It is only if Frep and Fatt forc es of a very stiff c harac ter are posited that we will witness a lattic e behavior similar to our pendulum ball expec tations. In other c irc umstanc es; the blow at pR might induc e negligible transmissive effec ts at pL (e.g., we might see a point-mass simulac rum for a line of pendulums c omposed of putty). Page 15 of 45

What is “Classical Mechanics” Anyway? All of these spec ific requirements upon Frep and Fatt fall under our earlier heading of “c onstitutive modeling c onditions.” How did we manage to overlook suc h constitutive concerns in our original reasoning about our pendulums? The answer is that we inadvertently punned on the term “forc e balanc e,” thereby lifting a loc al point- mass requirement on Frep and Fatt to the level of visible “balls.” The shift is fac ilitated by the innoc ent-looking invoc ation of a distinc tion between “external” and “internal” forc es, where it appears as if all of the “internal forc es” in the pac k possess their required “third law” c orrelates, while the leftward forc e on bR lac ks a matc h, a lapse that bR c an only rec tify through its “inertial reac tion.” There is a c elebrated passage in Thomson and Tait that explic itly interprets Newton's third law in this “lifted” manner: [I]f we c onsider any one material point of a system, its reac tion against ac c eleration must be equal and opposite to the resultant of the forc es whic h that point experienc es, whether by the ac tions of other parts of the system upon it, or by the influenc e of matter not belonging to the system. In other words, it must be in equilibrium with those forc es. Henc e, by the princ iple of superposition of forc es in equilibrium, all the forc es ac ting upon the system form, with the reac tions against ac c eleration, an equilibrating set of forc es upon the whole system. This is the c elebrated princ iple first explic itly stated, and very usefully applied, by d'Alembert in 1742, and still known by his name.23 But if we do that, we abandon some of the original spec ific s that permit a ready pathway from Newton's three laws as we interpreted them to the c onservation of energy and the like. What did Newton himself intend by his “third law”? His examples suggest drifts in his own thinking, sometimes straying c lose to those of Thomson and Tait. As indic ated earlier, most physic ists had firmly abandoned the point-mass approac h by 1850 or so, only to be revived in the twentieth c entury as offering the easiest pedagogic al bridge to quantum theory. Why did this happen? A number of salient c onsiderations c an be extrac ted from the wonderful artic les that James Clerk Maxwell c omposed for the c elebrated ninth edition of the Encyclopedia Britannica.2 4 Many of his c onc erns trac e to the simple fac t that natural materials vibrate in the manner that spec trosc opy indic ates and c an transmit waves. But attempts to c onstruc t point-mass lattic es c apable of imitating the experimentally determined behaviors usually proved disappointing, whereas models c onstruc ted upon c ontinuum or rigid body princ iples did muc h better. For example, in the 1820s Claude-Louis Navier had developed a c elebrated point-mass model for elastic materials leading to substanc es whose mac rosc opic behaviors are c harac teriz ed entirely by their Young's modulus. Working from general princ iples in a top-down, c ontinuum mec hanic s mode, Cauc hy instead c onc luded that isotropic elastic materials require two independent c onstants (Poisson's ratio in addition to Young's modulus) to fix their behaviors rather than Navier's solitary value. These issues were of great sc ientific moment bec ause the varieties of wave that c an travel through an elastic material are intimately linked to these c onstants. After a long period of c ontroversy, Cauc hy's “multi-c onstant” predic tions were eventually c onfirmed by experiment. By the end of the c entury, it was widely presumed that nature was c omposed of c ontinua of some sort, with its apparent point-like “partic les” c omprising whirlpool-like struc tures within an underlying c ontinuous medium.2 5 Cauc hy did not fully apprec iate the methodologic al advantages of the approac h he initiated (he sometimes worked in Navier's bottom-up mode as well), but later writers suc h as Green and Stokes strongly emphasiz ed the merits of the top-down approac h, whic h eventually bec ame the c ore c onstruc tion within modern c ontinuum mec hanic s (in a manner we shall survey in sec tion 5). To this day, their top-down tec hniques generally supply more reliable models with respec t to the materials of mac rosc opic experienc e. In fac t, many of the c elebrated philosophic al perc epts developed by writers suc h as Pierre Duhem and Ernst Mac h in the late nineteenth c entury trac e, in part, to their apprec iation of the desc riptive superiority of the top-down methods. More rec ently, the rise of swift c omputers has rendered the projec t of working direc tly with point-mass swarms in a bottom-up manner a more viable enterprise, but the results obtained are generally more suggestive than ac c urate. Shortly after Cauc hy's work, Poisson was able to reproduc e the “two c onstants” predic tions from a molec ular model c omposed of attrac ting spheroids rather than point masses. Likewise, one obtains better results within molec ular simulation today by working with swarms of extended bodies rather than points, although the c omputational c osts are muc h higher. But, from a foundational point of view, these modeling adjustments transport us into the realms of rigid body mechanics, whic h we shall c anvass in the next sec tion. Some folks, however, bec ome so smitten with point masses that they strive mightily to found “c lassic al mec hanic s” Page 16 of 45

What is “Classical Mechanics” Anyway? upon that basis, no matter how physic ally implausible the c onstruc tions they employ may appear. Thus, we might theoretic ally piggybac k upon Poisson's “two c onstant” results by c ollec ting large swarms of point masses into moc k spheroids held together by strange attrac tive forc es. But suc h assemblies bear no relationship to any structures present in real-life materials (whereas Poisson's spheroids often do). I am not sure what one gains from vain reductive enterprises like this. A logic al observation is pertinent as well. When one strives to explain why modeling princ iples P* work well at sc ale level ΔL* based upon the princ iples P operative at ΔL, one is further obliged to explic ate why the P* princ iples operate over the full range that they do. It is often easy to c onstruc t spec ific “toy models” at a ΔL level that will implement the desired P* behaviors at the ΔL* sc ale, but one little skirmish does not win a war. At best, one has merely built what the Vic torians c alled a P- princ iple analogy to the P* events. To be sure, the c onstruc tion ensures that some of P* 's ontologic al c laims are tec hnic ally c ompatible with P, but this signifies c omparatively little if the supportive “analogies” require suc h elaborate c ontrivanc es on a ΔL sc ale that they c annot serve as general underpinnings for the higher sc ale behaviors. I would have presumed that this observation was so obvious that it is sc arc ely worth drawing, but several times in the past year I have heard philosophers proudly dec lare that they have “derived the Navier-Stokes equations” (or the like) upon a more elementary basis, when, in fac t, they had merely c onc oc ted a weak and c ontrived analogy to suc h a system (by suc h standards, one c an probably “found” the same equations upon The Pickwick Papers). There are many loose c laims afloat within the philosophic al world as to how the various branc hes of physic s allegedly “reduc e” to eac h other; readers should approac h most of these with a wary eye. There is a final issue we should survey before returning to our main themes. As I have explic ated our Eulerian rec ipe, it fails as a modeling sc heme as soon as quantities like ac c eleration lose their required features. But this is exactly what happens if, for example, a point mass runs into another point mass or into one of the hard-shell barriers disc ussed earlier. From a stric t point-mass perspec tive, one should not tolerate ac c eleration-destroying interactions. But fulfilling this ambition in a plausible manner is not easy (and we must furthermore tame the additional blowup problems that emerge in the Xia c onstruc tion mentioned above). In real-life modeling prac tic e, “impac tive” enc ounters between point masses are usually addressed through ad hoc remedies that temporarily relax our Euler's rec ipe requirements, rather than searc hing for elusive spec ial forc e laws. In fac t, Newton's own approac h to billiard c ollisions implements this basic “turn off the laws temporarily” stratagem. He surrounds the c enter of eac h ball with a c risp finite boundary (so that the c entral mass point is c redited with a “hard shell potential,” although utiliz ing that voc abulary is quite anac hronistic in applic ation to Newton) (figure 2.12). Figure 2.12 Whenever these radii c ontac t one another (we shall only worry about the head-on c ollision c ase), Newton abandons the requirement that the “a” in “F = ma” must make sense and shifts his foc us to the two balls’ inc oming stores of linear momentum and kinetic energy (as we now dub them), together with a purely empiric al fac tor c alled a coefficient of restitution (it governs how muc h the total kinetic energy budget will diminish post-c ollision). In effec t, this treatment bloc ks out the c ruc ial interval of time Δt where “F = ma” fails to make sense and glues together the inc oming and outgoing events exterior to Δt through a mixture of c onservation princ iples2 6 (c onservation of linear momentum) and raw empiric s (c oeffic ients of restitution extrac ted from experiment). Formally, tac tic s that patc h over problematic intervals or regions in this manner are frequently c alled matched asymptotics. It is now time to extrac t the c entral morals of our disc ussion from the underbrush of spec ific s. The main desc riptive holes within the point-mass approac h trac e to the absenc e of the spec ial forc e laws that would be needed to Page 17 of 45

What is “Classical Mechanics” Anyway? c omplete its Eulerian rec ipe for ODE model c onstruc tion. It is hard to repair these lapses with any assuranc e bec ause the missing laws c onc ern the nature of c lose range c ohesive forc es and nature offers few robust indic ations as to how a c lassic al point mass modeler should tac kle suc h phenomena. In c onsequenc e, our underlying rec ipe lac ks many of the ingredients it would require before it c ould ratify, upon a purist point-mass basis, the many non-punc tiform modeling tec hniques that prac titioners regularly employ at higher ΔL* sc ale lengths. In pedagogic al prac tic e, these lapses in c onstitutive modeling are frequently disguised by c overt lifts to alternative approac hes better suited to the ΔL* level tec hniques. But c onc eptual c omplic ations within those rival sc hemes enc ourage frequent retreats bac k to the stolid redoubt of point masses, where the c onc eptual setting—if not the livin’ itself—is easy. The result is an intellec tual landsc ape poc kmarked with easy lifts and quic k esc apes that c an seem quite perplexing if your physic s instruc tor assures you that everything you see is rigorously wrought and intellec tually beyond reproac h. 4. Rigid Bo dy Mechanics Let us now investigate the foundational prospec ts for a physic s resting squarely upon a basic ontology c omposed of rigid bodies interac ting through c ontac t. There are a number of somewhat different treatments available in this arena, falling under the generic heading of “analytic al mec hanic s.” Our plan is no longer to analyz e suc h bodies as swarms of point masses or to allow them any internal flexibility. Ac c ordingly, when our ensembles of rigid parts flex, it must be through the internal realignment of c ompletely stiff c omponents, maintained as a c oherent c ollec tion through an admixture of ac tion-at-a-distanc e attrac tions and direc tly c ontac ting linkages suc h as hinges, pins, wires, and so on.2 7 The rigidity of any part is mathematic ally expressed by the fac t that its c urrent loc ation c an be c ompletely fixed by six numbers: three Cartesian c oordinates to loc ate a representative point within the body, and three angles to indic ate how the figure has rotated about that point. Any c onnec tion between suc h parts is usually c alled a constraint and is expressed with c onstraint equations that interrelate the c oordinates of the sundry parts.2 8 Two useful paradigms that I shall often c ite for the systems under review are (i) a bead sliding fric tionlessly along a rigid wire and (ii) the sewing mac hine mec hanism illustrated (figure 2.13). Figure 2.13 One does not expec t a normal lattic e of point masses to remain c ompletely rigid when disturbed—the gentlest attempt to move them en masse is likely to send little waves of disturbanc e ac ross the swarm. But if we c an safely assume that substantial hunks of a mec hanism remain approximately rigid in their gross movements, we c an potentially ignore a huge amount of internal c omplexity within the devic e. Consider our sewing devic e, whose bottom ec c entric link is turned by a motor. If we were forc ed to model these arrangements explic itly as a swarm of strongly attrac ting point masses, we would need to painstakingly plot how the binding forc es allow the input movement to gradually transmit itself from one little piec e to another ac ross the mec hanism. This story must surely involve very c omplex proc esses in light of the branc hing c ausal pathways that initiate at the motor. As observed earlier, it is sc arc ely evident that orthodox point-mass mec hanic s c ontains enough internal resourc es to provide an adequate simulac rum of the expec ted behaviors. But onc e we are assured that the c omponent piec es will remain nearly rigid throughout all of the devic e's ordeals, high sc hool geometry c an c ompute exac tly how muc h the needle will wiggle as the drum at the bottom gets turned through an angle θ. Admittedly, this is not a trivial high- Page 18 of 45

What is “Classical Mechanics” Anyway? sc hool c alc ulation, but its demands are vastly simpler than c omputing how the whole point-mass swarm will behave under the same c onditions. In other words, we c an employ our upper sc ale ΔL* knowledge that our sewing mac hine parts stay rigid and obey their c onnec tive c onstraints to avoid the very c omplic ated mec hanic al relationships that hold among the devic e's c omponent masses at the ΔL level. Or, at least, that is how our target mec hanism would appear from a pointmass perspective. But in the present sec tion, we wish to c onsider “foundations” for c lassic al mec hanic s in whic h notions like “rigid body” and “pinned c onstraint” c omprise the mechanical primitives of the subjec t and are not introduc ed as c onvenient approximations to c omplex point-mass underpinnings. Figure 2.14 Figure 2.15 This perspec tive supplies analytic al mec hanic s with a huge c omputational advantage over point-mass-based modelings. Thomson and Tait artic ulate these virtues as follows: “[T]he forc es whic h produc e, or tend to produc e, [the ac tions] may be left out of c onsideration. Thus we are enabled to investigate the ac tion of mac hinery supposed to c onsist of separate portions whose form and dimension are unalterable.”2 9 Earlier we noted that it is hard to model, from a point-mass perspec tive, the simplest forms of redirection of thrust, as oc c urs when a plug slides along a c urved trac k. In our sewing mac hine, the redirec tion is of a far c leverer design, but proc eeds ac c ording to the same analytic al mec hanic al princ iples. To formulate doc trines of this type c orrec tly, we generally need to c apture the system's c urrent c onfiguration in terms of generalized coordinates, rather than the Cartesian c oordinates that are c entral to the point-mass reading of the sec ond law. Often, good generaliz ed c oordinates use natural c oordinates of the kind mentioned earlier: quantitative measures of displac ement that are c losely c orrelated with the system's available motions. For a plug sliding in a slot, its plac ement in terms of arc length along the slot represents the single natural c oordinate we require to fix the plug's position, whereas its three loc ations expressed on Cartesian axes do not relate to the motion in any internally “natural” way (figure 2.14). A sec ond vital feature of the c oordinates usually employed in analytic al mec hanic s is that they are independently variable with respec t to eac h other: any spec ific c oordinate c an be altered without nec essarily disturbing the others. With a steam shovel, for example, we will want to employ the five dec ompositional movements illustrated to fix its c onfiguration, rather than the regular Cartesian c oordinates of its many parts. The latter are desc riptively Page 19 of 45

What is “Classical Mechanics” Anyway? entangled in a manner that prevents us from applying the usual forms of virtual-work reasoning (figure 2.15). Figure 2.16 Muc h of the prac tic al suc c ess of analytic mec hanic s trac es to the fac t that suitable independent c oordinates for a c omplex system c an often be divined simply through experimentally determining how it wiggles under manipulation and the direc tions in whic h input thrust travels ac ross its interior. Suc h data represents raw higher-sc ale information about our system's dominant behaviors. As indic ated earlier, as soon as material bodies genuinely fill finite volumes, a new type of “forc e” quietly enters the sc ene. This forc e eventually bec omes a sec ret sourc e of signific ant tensions within mec hanic al thinking. Sinc e point masses are inherently z ero-dimensional in nature, they c an be provided with a surrogate for normal “siz e” only by erec ting rough “effec tive volumes” through a battery of strong, short-range repulsive forc es. But if our fundamental objec ts possess true siz es, then the contact forces will arise upon the interfac e between two c ontac ting bodies. These new items c an no longer qualify as action-at-a-distance forces simply bec ause no distanc e separates the embedded points where the transmission oc c urs (figure 2.16).3 0 There are two grades of c ontac t forc es with whic h we must eventually deal. The first are the boundary forces applied along the outside surfac e ∂B3 1 of a body B, suc h as a loaded weight passively resting on top of a four bar mec hanism or a hammer blow applied somewhere (the first is traditionally c alled a static load or a dead load and the second a dynamic loading) (figure 2.17). These are the only c ontac t forc es at issue within rigid body mec hanic s. But when we turn to flexible c ontinua, a sec ond grade of interior c ontac t forc es emerge in the guise of the traction forces that appear ac ross the boundary of (almost) any internal surfac e S that we might mark out within the larger body B. Suc h internal surfac es S are commonly called free body diagrams or Eulerian cuts (figure 2.18). Figure 2.17 Page 20 of 45

What is “Classical Mechanics” Anyway? Figure 2.18 Eac h suc h S will bristle with an array of traction forces that point either inward or outward at eac h surfac e point—it is then presumed in third-law fashion that the material outside S will push or get pulled in the opposite direc tion at that same plac e. The most familiar exemplar of suc h trac tion vec tors are the normal pressures ac ting within a nonvisc ous fluid, but the c omplic ated internal pushes and pulls operative within other flexible bodies mandate the introduction of the more general notion of stress. But onc e the interior of a body is c laimed to be c ompletely rigid, as we shall assume throughout the present sec tion, then this interior grade of c ontac t forc e bec omes ill-defined, as do allied notions suc h as internal pressure. So we will not worry about how to deal with suc h internal trac tions now and will c onc entrate upon the surfac e forc es that appear along the boundaries between c ontac ting bodies. It is c ommon for elementary textbooks to vaguely c laim that all forms of c ontac t forc e really represent short-range c ohesive forc es between separate partic les. This c ontention might be true insofar as c ontac t forc es represent classical distillations of quantum processes of roughly that c harac ter, but suc h asseverations c an prove very misleading insofar as our text appears to be c onc erned with c lassic al proc esses exc lusively, where it is not evident that plausible short-range c ohesive forc es of a c lassic al point-mass c harac ter c an ground, on a ΔL basis, the standard rigid body or c ontinuum behaviors witnessed upon a ΔL* sc ale. In fac t, suc h offhanded appeals to short-range forc es often disguise the fac t that fundamentally new issues about how forc es operate mathematic ally appear on the sc ene as soon as c ontac t forc es are tolerated. Often their resolution requires that we reinterpret Newton's laws in a signific antly altered manner or turn to other forms of “foundational princ iple” altogether. In the fac e of these c onc eptual c hallenges, haz y appeals to fic titious short-range forc es between point c enters at a lower sc ale length ΔL merely serve as c onvenient esc ape hatc hes that allow authors to evade addressing these foundational issues squarely (these evasions bec ome partic ularly troubling in the c ontext of c ontinua, as we shall later see). To be sure, the texts eventually stagger their way to the requisite ΔL* level equations, but only along pathways that are apt to c onfuse a c ritic al student. One of these diffic ulties—whic h arises even with the exterior boundary forc es of rigid body mec hanic s—trac es to the simple fac t that there is a disparity in dimension between c ontac t forc es that ac t upon bounding surfaces ∂B and forc es suc h as gravity that ac t upon the loc aliz ed points inside B. As soon as our attention shifted to extended bodies, we should have properly stopped c alling suc h items “forc es” at all and instead c onsidered “forc e densities” of dimensionally inc ompatible grades. The motives for these adjustments trac e to the usual diffic ulties of making sense of c ontinuously distributed quantities that date to the time of Zeno. Page 21 of 45

What is “Classical Mechanics” Anyway? Figure 2.19 Suppose we have a target with a bull's-eye and two arc hers: skilled Marian and inept Robin (figure 2.19). What are Marian's and Robin's respec tive probabilities for hitting the exac t c enter of the target c? Answer: most likely z ero in both c ases, bec ause if the “hit c exac tly” answers were c redited with any finite amount ε, then (under the assumption that points near to c should be c redited with probabilities c lose to ε) the summed probabilities of hitting any finite region of the target will bec ome infinite (due to the infinity of individual points c ontained in suc h a region). But if Marian's and Robin's probabilities for hitting the target at any individual point are always z ero, shouldn't it follow that their summed probabilities of hitting any finite region A also need to be identic al (viz ., z ero), rendering them equally lousy marksmen? Obviously not, but the task of straightening out these riddles is the business of the modern theory of measure. This theory addresses our problem by c rediting Marian and Robin with different probability densities with respec t to the individual points in the target. To extrac t a proper probability from a density, one must “add up” (integrate over) these densities over suffic iently large areas. Based upon their different densities, the true probability differences between Marian and Robin's skills will show up only after suffic iently large expanses of target c ome into c onsideration. Getting all of this to work out c orrec tly requires very c areful mathematic al preparation. Plainly, we need to adopt similar polic ies with respec t to our new “forc es”: c onsidered at a point-length sc ale only the force densities c an be nonz ero—true forc es should not emerge until we have integrated these loc al densities over larger regions. Figure 2.20 The awkward tension that segregates surfac e forc es from body forc es suc h as gravity stems from the fac t that, c onsidered properly as densities, their respec tive quantities must be dimensionally inharmonious. Why? In the c ase of the trac tions pulling and pushing upon a boundary ∂B, we expec t to reac h genuine resultant forc es after we have integrated over finite stretc hes ∂S of the exterior surfac e ∂B. But with gravitation, we must integrate over volumes V of points within B itself (not just along stretc hes of ∂B) before we c an assemble forc es of c omparable strength from gravitational attrac tion (figure 2.20). Considered from the point of view of the normal volume measure on B, any surfac e piec e ∂S will qualify as “of 0 measure,” so we c annot use this same measure in dealing with c ontac t forc es. In sum: genuine forc es c an be assembled from muc h smaller sets of points in the c ase of a c ontac t forc e than in the c ase of gravity, for they reac h the level of a “finite resultant forc e” more quic kly in the former c ase. Page 22 of 45

What is “Classical Mechanics” Anyway? It is only after these surfac e and volume resultants (the fat arrows in the diagram) have been obtained that we will possess genuine forc es—not densities—that c an be meaningfully combined. This dimensional disparity of our densities is not merely an awkward mathematic al issue, for the fac t that surfac e forc es inherently overwhelm body forc es within small regions plays a vital role in determining the logic al c harac ter of vital notions like “stress.” We shall disc uss these features in sec tion 5. Let us return to the problem of c ombining body and surfac e forc es, now c onstrued as densities. We find that two basic giz mos are needed to fulfill the roles that “total forc e” serves within point-mass mec hanic s. We first require a dimensionally c orrec t analog for the notion of total forc e, whic h we now c ompute as the vec tor resultant of two density integrations ∫S fs ds and ∫ V fb dv (where fs and fb are the surfac e forc e and body forc e densities, respec tively). Observe that these two integrations transpire over the requisite regions: S for outer surfac e and V for interior volume. In so doing, we are summing a large number of forc e densities that ac t in different loc ales, unlike in the point-mass c ase where forc es all ac t in the same plac e. But in c omposing our new notion of total forc e, we simply ignore these differenc es in point of applic ation. Using these new notions, we obtain an analog of Newton's Sec ond Law suitable to isolated rigid bodies: (∫V ρ dv) d2ri/dt2 = ∫S fs ds +∫V fb dv, where ∫ V ρ dv is the summation of the mass density over the entire rigid body B. But with whic h point in B should the loc ation r in the term (∫V ρ dv) d2 ri/dt2 be c omputed? It does not really matter: every point will display the same linear ac c eleration in any direc tion we look. Some writers link r to the c enter of mass of the body, but there is no espec ial reason for doing so (espec ially when the c enter of mass is often not located inside B at all, as in a doughnut). Although the points in B ac c elerate in the same way, they c ertainly do not have the same veloc ities. An additional notion is needed to desc ribe the veloc ity relations, or the turning of a rigid body. This new notion is c alled the torque τ (or turning moment) of the summed forc e densities ac ting upon B. Onc e again, this summation needs to be broken into two integrals that separately average the lever arm c ontributions of the surfac e and body forc es with respec t to some c enter A within B (it does not matter where, although c ertain c entroids c an make the c alc ulations easier) (figure 2.21). More exac tly, τ = ∫S (fs × r)ds +∫V (fb ×r) dv, where r represents distanc e to the c hosen referenc e point A. Quite different distributions of forc e density ac ross a rigid body c an move it in identic al ways as long as their averaged total forc e and averaged torque about A are the same. Figure 2.21 To c omplete our sc heme, we must now quantify how our rigid body c reates an inertial resistanc e to an applied torque as well. Here we need to c ompute how far away from A the mass density ρ within B tends to lie on average (viz . ∫ V (ρ. r2 ) dv). This new quantity I is c alled B's “moment of inertia” around A. Using it, we c an express “Euler's Sec ond Law of Motion”3 2 for torques as I d2 θ/dt2 = ∫S (fs× r)ds+∫V (fb × r) dv, where d2 θ/dt2 is the angular acceleration of B. Working within a point-mass framework, Euler's sec ond law is provable from Newton's sec ond law in c onjunc tion with the third law restric tions on ac tion-at a-distanc e forc es. But this dependenc e no longer holds as soon as the forc es tolerated multiply into new varieties. In partic ular, Euler's sec ond law is required as an independent postulate to show that stress tensors must be symmetric within a c ontinuum physic s setting. Unjustified lifts from point-mass mec hanic s often disguise this c ruc ial fac t in many texts. Page 23 of 45

What is “Classical Mechanics” Anyway? However, our two Eulerian princ iples alone do not tell us how hinged assemblies of rigid bodies should ac t, whic h is our main objec tive in this sec tion. A general answer to this question was supplied by Lagrange, who elevated some of the reasonings we have already c anvassed into a general framing princ iple. Spec ific ally, Lagrange maintained that, in any system of rigid parts c harac teriz ed by n sites of impressed forc e, either (i) the devic e remains in equilibrium and the total virtual work assoc iated with all impressed forc es vanishes or (ii) the devic e moves with exac tly the requisite inertial reac tions at the n sites to c ompensate for the virtual work imbalanc e. Traditionally, c onsideration (i) is dubbed the principle of virtual work and (ii) is c alled d'Alembert's principle. Combined into one formula, we obtain Lagrange's principle: (1) where δqi represents a “virtual adjustment” in the c oordinate value qi, leading to a measure of the work that the applied forc e Fi would supply if it c ould be prolonged through that distanc e.3 3 Lagrange's formula is partic ularly useful if we have employed independent c oordinates as our qi, for then we c an write down a formula that expresses how work supplied at, for example, site q1 gets transmitted ac ross the mec hanism to any other site on the assumption that the remaining sites can stay fixed in the proc ess. Working out these rules for eac h pair of sites provides a c ollec tion of equations that c an c ompletely fix how our hinged rigid body will move. The formulas familiar from analytic mec hanic s that are c ouc hed in terms of “Lagrangians” or “Hamiltonians” simply represent these new equations rewritten reliant upon c ertain further assumptions about the nature of the forc es at issue (viz ., their derivability from potentials). Many interesting geometric al problems are c losely c onnec ted to the generaliz ed-c oordinate representations of rigid-body mec hanic s. The c onfiguration spac es of our earlier point-mass swarms are c omparatively uninteresting from a mathematic al point of view. But if one c onsiders the mobility space of a c omplic ated mec hanism like our c rane, as determined by its varying generaliz ed c oordinates, we obtain a quite novel struc ture, largely bec ause its c oordinates are angle-like in c harac ter: they return to their starting values after a 360° rotation. Mathematic ally, we obtain suc h mobility spac es by c utting out all of the “c an't be visited” regions from a regular Cartesian 3n spac e and gluing together the remaining edges ac c ording to the pathways of angle-like returns. The resulting substruc tures c an prove very c omplic ated geometric ally and c omprise a topic of great mathematic al interest far beyond the limits of the kinematics of mechanisms (whic h is the traditional name for the study of mac hine mobility). Figure 2.22 Figure 2.23 From a point-mass vantage, we are plainly skipping over a huge amount of internal struc ture. Let us examine a small piec e of our c rane from a punc tiform point of view (figure 2.22). Clearly, strong c ohesive forc es Fij will be required to loc k point i into a lattic e with point mass j and some kind of binding forc e FC will also be needed to keep our piec e fixed to its pin. All mention of these has vanished from Page 24 of 45

What is “Classical Mechanics” Anyway? C Lagrange's princ iple. Why are we allowed to ignore these extra forc es? Textbooks c ommonly argue as follows: (1) “The net work of the c ohesive forc es vanishes bec ause they oc c ur in internal-forc e pairs where Fij = −Fji. Sinc e their virtual displac ements will be the same, their virtual work c ontributions will c anc el eac h other out.” Or: (2) “The c onstraint forc e Fc does no work bec ause its ac tion is orthogonal to the path of virtual movement δrc .” Here is Donald Greenwood's version of this last argument, presented in the c ourse of “justifying” Lagrange's princ iple from a point-mass standpoint: “[C]onsider a body B whic h slides without fric tion on a fixed surfac e S…. The c onstraint forc e is normal to the surfac e at the tangent point P, but any virtual displac ement of P involves sliding in the tangent plane at that point. Henc e no work is done by the c onstraint forc e R in a virtual displac ement” (figure 2.23).34 Figure 2.24 But in our point-mass frame, all forc es are supposed to ac t from one point to another along the line c onnec ting them. But our c onstraint forc e R looks as if it starts and ends in exac tly the same spot P, whic h was not permitted under our old reading of the third law. Plainly, some new kind of “forc e” has been smuggled into Greenwood's text, without adequate prior warning. On virtually the same page Greenwood argues for Lagrange's princ iple in a different setting as follows (figure 2.24): Assume that two partic les are c onnec ted by a rigid, massless rod…. Bec ause of Newton's third law, the forc es exerted by the rod on the partic les m1 and m2 are equal, opposite and c ollinear. Henc e R2 = −R1 … as shown. Furthermore, sinc e the rod is rigid, the displac ement c omponents in the direc tion of the rod must be equal or e.δr1 = e.δr2 [where e is a unit vec tor pointing in the direc tion of the rod]. Therefore the virtual work of the c onstraint forc es is z ero: δW = R1.δr1+ R2 .δr2 = 0.3 5 But by what right c an we insert a “rigid, massless rod” in our system and still maintain that “Newton's third law” equates R2 = −R1? It is not as if the two masses are direc tly exerting forc es on one another, as our earlier reading of the third law expec ts. Indeed, suppose that the intervening rod is c urved, rather than straight. We still want our reasoning to hold, yet plainly R2 ≠ −R1 (the vec tors point in quite different direc tions). Passages like these trade upon unnotic ed elisions between the foundational sense of “isolated partic le” and looser policies of talking of “representative points” within more extended bodies. By exploiting our alleged freedom to place representative points where we wish, Greenwood allows point masses to sometimes sit on top of one another or loc ate themselves at the far ends of ghostly, massless rods. Through simple appeals of this c harac ter, we find ourselves mirac ulously lifted to the c harac teristic sc ale level of objec ts far above the realm of the c omponent partic les (atoms, molec ules, the tiny c rystals in iron) that we originally sought to model as point masses. Observe that a doc trine that is essentially philosophic al in nature (“sc ientists idealiz e their targets through selec ting representative points”) has been tac itly employed as a c over for a missing stretc h of substantial mathematic s (“how do point-mass foundations support the princ iples of analytic al mec hanic s?”). As we observed, one of Hilbert's stated objec tives in his sixth problem was to study these lifts in a more rigorous spirit, although he did not observe that stoc k textbook arguments like Greenwood's involve moves of this c harac ter. We also disc ussed the general manner in whic h constitutive modeling considerations get mysteriously bypassed in arguments of this c harac ter. But in the situations Greenwood disc usses, we have somehow persuaded ourselves that we c an derive salient predic tions based upon “general laws” pertinent to point masses alone, without needing to supply any c onstitutive modeling that c an explain why systems would behave differently if they had been composed of nonrigid parts. Thus, standard “derivations” of analytic al mec hanic s doc trines from point-mass foundations are rarely c ogent, if sc rutiniz ed from the point of view of Hilbertian rigor. But none of our c onsiderations show that an alternative set of foundational princ iples c annot be c oherently framed that ac c epts rigid bodies as its primitive objec ts, possibly in c onjunc tion with point masses as well. In fac t, the best modern writers on mec hanic s rec ogniz e that pretending that Page 25 of 45

What is “Classical Mechanics” Anyway? analytic al mec hanic s c an be adequately founded upon point-mass foundations is simply a mistake. Cornelius Lanc z os c omments: Those sc ientists who c laim that analytic al mec hanic s is nothing but a mathematic ally different formulation of the laws of Newton must assume that Lagrange's princ iple is deduc ible from the Newtonian laws of motion. The author is unable to see how this c an be done. Certainly the third law of motion, “ac tion equals reac tion,” is not wide enough to replac e Lagrange's princ iple.3 6 He partic ularly has in mind some of the third-law ambiguities disc ussed in sec tion 2. In c ritic iz ing derivations like Greenwood's for their lac k of rigor, we should never forget that the modeling tec hniques they are intended to justify are of vital importanc e to working physic s. For the import of virtual-work sc hemes in prac tic e is that they allow us to avoid working through an awful lot of diffic ult physic s that runs the risk of introduc ing large errors into our c alc ulations with little gain in predic tive power. We have already disc ussed the advantages of working with data drawn from a range of sc ale siz es. If we already know how the princ ipal patterns of thrust transmission operate within our c rane at a large sc ale siz e ΔL*, why not exploit that information to reduc e the c omplexity of our modeling, even at the c ost of a c ertain degree of approximation with respec t to the point masses that c omprise it at a sc ale ΔL* The essential genius of virtual work and the other tec hniques of analytic al mec hanic s lies in their ability to c ombine data types in this manner. Analytic al mec hanic s, if stoutly set on its own feet axiomatic ally, should appear an odd c hoic e for serving as a baseline ontology for c lassic al mec hanic s due to the tremendous number of desc riptive holes it c ontains. This sec tion has devoted its attention to analytic mec hanic s’ prospec ts as a foundational enterprise largely bec ause the subjec t c ommonly serves as a favored point of refuge when one enc ounters c onc eptual diffic ulties in pursuing our other basic ontologies. In fac t, the position of rigid bodies within c lassic al physic s is muc h like that of the disreputable unc le who possesses most of the money in the family: you do not fully admire his c harac ter but you apprec iate all of the good things he c an buy for you. We have already examined several ways in whic h point-mass mec hanic s c ommonly appeals to rigid-body notions as esc ape hatc hes when it finds itself in desc riptive hot water. Thus, we invoke “massless rods” to hide the fac t that we do not know the spec ial forc e laws that bind two point-masses at a c onstant distanc e. Or we enlarge our point-mass planets to bec ome finite spheroids to avoid Xia-type blowups. Or, like Poisson, we c orrec t the one- elastic -c onstant defic ienc ies of a material modeling by replac ing the point c enters with rigid ellipsoids. But these doors of c onc eptual esc ape swing both ways, for analytic al mec hanic s c ommonly evokes the resourc es of its ontological rivals to sustain its own reasonableness. 5. Co ntinuum Mechanics Figure 2.25 If we c ould mark out the salient differenc es c learly and poll most of the prominent c lassic al physic ists of the past with respec t to their favored c hoic e of foundational objec t, the majority would undoubtedly selec t c ontinuous, flexible bodies. Let us now try to artic ulate princ iples c apable of governing the behaviors of c ontinua c oherently. Page 26 of 45

What is “Classical Mechanics” Anyway? We immediately c onfront a more diffic ult form of the surfac e/volume forc e c oordination problem than we c onfronted in the c ase of rigid bodies. In rigid-body mec hanic s, the relevant trac tion forc es operate only along the exterior surfac es ∂B of the rigid body B under c onsideration. But inside a flexible body, we c an c arve out infinitely many internal surfac es ∂S able to support their own arrays of trac tion vec tors as well (figure 2.25). Furthermore, the trac tions on eac h different ∂S will generally differ from one another and from the exterior trac tions applied along the outer boundary ∂B. Indeed, we antic ipate that as we push and pull upon ∂B in different ways, these exterior modifications will make themselves felt at an interior point q through progressively altering the trac tion forc es upon all of the interior surfac es ∂S that surround q. Moreover, this proc ess of inward transmission will require some time to c omplete: the trac tions on ∂S* must alter before the trac tions on ∂S c an c hange. Inside a truly rigid body, however, such inner tractions and waves no longer make sense, for essentially the same reasons that the notion of “absolute pressure” bec omes problematic when a fluid is assumed to be inc ompressible. Usually, notions of “rigid body” are regarded as inc ompatible with a c ontinuum physic s point of view. In the previous section, we summed the surface forces around the outer boundary ∂B and the volume forces inside B employing two integrations whose results we then added to get a resultant applied force. We then learned that we should also c ompute a c ombined torque in a similar manner. With those two ingredients in plac e, Euler's two laws of motion c ould tell us how our rigid body would respond. But in that c ontext we only had to c ontend with the body forc es and trac tion vec tors around the outer boundary ∂B. How should we address the vast army of differing ∂S' s that have now entered our stage in the entourage of flexible bodies? If we naively c ompute resultant forc es and torques from these, we can obtain substantially different answers according to the inner surface ∂S chosen. This is a surfac e/body forc e c oordination problem of c onsiderably greater subtlety than we addressed earlier. The eventual solution invokes the notions of “stress” and “strain.” Before proc eeding further, a few words of warning are in order concerning these innocent-looking words. Most philosophers interested in physics have already run ac ross those words—in the guise, say, of their c lose c ousin, the “stress-energy tensor” of general relativity—without reflec ting suffic iently on their c onc eptual oddities (“stress” is not “just a form of forc e” and “strain” is not “just a form of shape c hange”). Historic ally, it was not until the end of the nineteenth c entury that the true novelty of these c onstruc tions was adequately rec ogniz ed.3 7 Some of this c onfusion trac ed to the c arelessness about “points” that “representative point” talk enc ourages. So let us reiterate that in dealing with interior points like q, we are not longer considering the isolated points of mass point physics: our new points come densely surrounded by infinitely many neighbors, situated as close to them as one might like. And they should not to be identified with points in the ambient c ontainer spac e; our material points (their most c ommon name) wander through that bac kground spac e in a trac kable way. It is best, at this preliminary stage of our disc ussion, to c onc eptualiz e our material points q, not as bare geometric al entities, but as decorated points or physical Figure 2.26 Figure 2.27 Page 27 of 45

What is “Classical Mechanics” Anyway? infinitesimals that have temporarily parked themselves at various spatial points p (which are not “decorated”). In partic ular, if q forms part of a solid material like iron and the material in q's immediate neighborhood is fully relaxed, we should pic ture its “dec orated” c ondition as an infinitesimal little c ube about the c entral point p (figure 2.26). But if q is subjec t to stress, infinitesimal trac tion vec tors should appear on eac h of its fac es, one to a side and pointing inward or outward in any direction desired. In response to these tractions, our little boxes will adjust their decoration, by adjusting their infinitesimal volumes or shifting shape in a shearing pattern (or displaying com- binations of these two basic alterations). In fact, the descriptive purpose of a stress tensor is to capture the local pushes and pulls of the traction vectors on q, while the usual (Cauchy-Green) strain tensor captures q's degree of distortion from its cubic relaxed state. Considering our material over a broader scale, we realize that the material points q found a*t locations near to q must be decorated in a manner very close to, but not identical with, that of q —otherwise, the material would display fissures (this relationship among nearby infinitesimals is c alled compatibility38) (figure 2.27). Bec ause all of these modes of dec oration oc c ur on an infinitesimal sc ale, stresses and strains behave like the densities introduc ed earlier in c onnec tion with mass and force: suc h measures do not sum to bec ome legitimate masses and forc es until we c onsider finite volumes of material. But the rules whereby loc aliz ed stresses and strains eventually sum to produc e finite c harac teristic s of the material stuff to whic h they belong are more c omplic ated than the proc edures used for simple densities. I have highlighted the odd, dec orated-point aspec ts of the material points of c ontinuum mec hanic s to help readers properly apprec iate the c onc eptual novelties that lie before us—we are no longer c onsidering the familiar isolated points of the easy-to-comprehend point-mass framework. And although we shall eventually appeal to various limiting proc edures to persuade our stresses and strains to work together harmoniously, readers should not presume that the conceptual difficulties of continuum physics are merely matters of “explaining away infinitesimals” in the familiar δ/ε fashion of elementary c alc ulus c ourses. No: deep questions of physical principle are c entral to our c onc erns; we are not simply striving to make hygienic sense of infinitesimal points. Our foundational difficulties might be fairly dubbed the problem of the physical infinitesimal, but our problems mainly belong to physics and are considerably more substantive than the comparable problem of the mathematical in- finitesimal from freshman calculus. To be sure, philosophers have sometimes presumed otherwise, but only as a result of underestimating the pertinent physic al c onc erns. It strikes me that many of the deepest worries about matter in our philosophic al heritage trac e, in one way or another, to our “problem of the physic al infinitesimal.” Figure 2.28 With these warnings not to underestimate stress and strain in hand, let us now turn to the foundational ailments for whic h they will eventually provide part of the c ure. Let us rec all the rather c omplic ated physic al situation that pertains at the level of the complete blob B of material to which q belongs (figure 2.28). Its interior will be pulled upon by gravity g and other action-at-a-distance forces of that ilk. But B will also be affec ted by the various twists and pulls that we exert direc tly upon its exterior surfac e ∂B as “c ontac t forc es.” If the material inside B is perfec tly rigid, the basic problem of c oordinating these two c lasses of “forc e” c ould be resolved fairly easily by computing resultant torques and applying Euler's two laws of motion for rigid bodies. But * Page 28 of 45

What is “Classical Mechanics” Anyway? this simple polic y works only bec ause the material is rigid: every point q* inside B must display the same linear ac c eleration and the whole will rotate in exac tly the same way no matter from whic h referenc e point its torque is gauged. However, if the matter inside B is not perfectly stiff (let B be a blob of jelly or water), then the response behavior immediately around q will usually look quite different from the c orresponding behaviors around q* (neither the loc al ac c elerations nor rotations will be the same). And it is these loc al differenc es within flexible bodies that allow them to c arry interior waves, whic h rigid bodies c annot support. When we twist and pull upon the external surfac e of a flexible body B, we generate a lot of internal trac tions, for the effec ts of our surfac e manipulations generate c ompression waves whose effec ts eventually reac h q by progressively altering the loc al trac tions upon a c ontrac ting c ollec tion of surfac es S1, S2 , S3 , … surrounding q. It is these internal surfac es and their shifting arrays of trac tion vec tors that greatly c omplic ate our earlier surfac e/body forc e c oordination problem in the c ase of flexible bodies. That diffic ulty, the reader will remember, trac es to the dimensional disparity generated by the fac t that c ontac t trac tions represent surfac e “forc es” (properly forc e densities) in the sense they must attac h to some shell of surfac e ∂S surrounding a point q before they c an be c oherently integrated, whereas body “forc es” (again, densities) suc h as gravitation apply direc tly to simple points q and need to be integrated over volumes. In the rigid- body c ase, only the outer layer of exterior trac tion pulls needed to c oordinate with its interior points q, but in flexible bodies we are c onfronted with a host of additional shells ∂S to c oordinate, appearing as the interior c uts whose c harac teristic s are c ontinually altered by the waves that pass through them. Figure 2.29 Plainly this represents a fairly complicated physical problematic. It is often remarked that physics is simpler in the small, indicating that uncomplicated laws of material behavior can be elegantly formulated only at the infinitesimal level. Will this methodology help us here? Consider the material at a point q, wher*e it will display a local mass density ρ and allied characteristics like charge. It will be pulled upon directly by gravit2ation a2nd other possible long- distance “body forces,” which can be summed to supply a local resultant vector g . We can presume our material will react to its full schedule of local pushes and pulls by manifesting an acceleration D q / Dt (the capital D's signify the material derivative, which is explained in every textbook on continuum mechanics). Unfortunately, the compression waves passing through q will also affect its full schedule of local pushes and pulls and it is these that make our force-coordination problem so difficult. It is easy to understand how a passing wave will affect a shell of surface ∂S: run a tangent plane through any point on ∂S and see which way the traction T supplied by the wave locally points across the plane. So to understand how the compres*sion waves will affect q, we should set up a little s*hell around q and compute the traction vectors on ∂S created by the passing waves. All we need to do, it would seem, is to co*mpute how the resulting surface “force” summation F should compare to the body force summation g * acting at q. But wait a minute: no part of ∂S is actually located at q and, in fact, we can easily carve out *a* smaller c*ut ∂S inside ∂S whose integrated tractions may differ considerably from those on ∂S itself (why? because ∂S is affected by different wave movements than ∂S) (figure 2.29). And we can draw an even smaller cut ∂S inside ∂S where the same phenomenon reappears. And so on, ad infinitum. In short, we have gone smaller in our physics, but nothing has become simpler! The regress traces, of course, to the fundamental scale invariance of homogeneous classical continua. Whatever characteristic length ΔL we choose, volumes of such materials will always behave exactly alike in terms of the principles they obey (of course, one can also consider composite continua where various sectors obey different rules, but these raise further difficulties, which we shall discuss later). Somehow we must arrest this regress of unprofitable descent if we hope to get anywhere in continuum physics. But how can we do this? One cannot blithely say, “Oh, just take a ‘limit’ as you shrink to q,” for it is not at all apparent what should happen to our traction forces when the cuts on which they live shrink to nothingness at q itself. (1) Will the result be merely a simple pressure, which operates to expand or contract our element in terms of its volume? Page 29 of 45

What is “Classical Mechanics” Anyway? (2) Can suc h loc al “pressures” pull differently in different direc tions? (3) Can the direc tionalities of our trac tions lean sideways in a manner that c an shear an infinitesimal blob S without altering its volume? (4) If so, will they ac t differently upon different planes around S? (5) How differently? (6) If so, how muc h latitude c an they display with respect to these variations? (7) Will turning torques also leave a residual infinitesimal turning moment within S? Figure 2.30 The standard (although not invariable) answers to these questions are: (1) no; (2) no; (3) no; (4) yes; (5) yes; (6) they must interrelate in the manner of a 3D vec tor spac e; (7) no. But few of these should seem entirely obvious. Internal pressures, for example, c an vary c onsiderably ac ross a fluid—mightn't these longer range inequalities deposit an unbalanc ed pulling upon our small blob S as a loc al residue? Prior to the time of Cauc hy, the greatest prac titioners of mec hanic s answered “no” to (3), often on the basis of the way in whic h they c orrec tly answered “no” to (7).3 9 Although the c onventional textbook response to (7) is “no,” there are well-developed theories of direc ted media that address this question differently. The fac t that it is hard to augur intuitively how infinitesimal portions of a c ontinuous medium should behave helps explain why the old c ontroversy between rari- and multi- c onstant theories of elastic ity took so long to resolve. Suc h questions c onc ern only the static responses of materials. Onc e dynamic s c ome into play, an even wider range of difficult questions emerge. Can our infinitesimal elements retain long-term “memories” of their previous history? Certainly, macroscopic media often behave in this way: two identic al looking paper c lips made of the same material may respond differently to bending pressures bec ause c lip A has been flexed many times in the past but c lip B has not. Can an infinitesimal blob S display allied memories as well, or must suc h proc esses emerge due to c omplic ated interac tions between finite portions of a c omposite system? Likewise, might our “infinitesimals” display “delayed memories” in the sense that a blob S might respond to altered c onditions in a non-immediate manner? Again, toothpaste ac ts like this: it gradually “remembers” its shape bac k in the tube and tardily reverts bac k to it (figure 2.30). Suc h questions lay behind the twentieth-c entury revival of interest in the “foundations” of c lassic al c ontinuum mec hanic s: sc ientists who c onfronted with new industrial substanc es needed guidance as to how suc h complicated materials might be reliably modeled. Ultimately, the answers to all of these questions depend upon physic s bec ause, insofar as mathematic s is alone c onc erned, suc h issues c an be resolved in many different ways. That is why our “problem of the physic al infinitesimal” (whic h is equivalent to answering our questions c oherently) is not mainly an issue in δ/ε rigoriz ation. In sum, we are c onfronted with a serious c onc eptual regress: the c omplic ated behaviors of these materials never seem to bec ome simpler no matter how small the portions we c onsider. How c an we halt this unhelpful desc ent into what Leibniz c alled “the labyrinth of the c ontinuum”? I will first sketc h two traditional answers and then the modern view. The first of these c laims that at some minute sc ale length ΔL, the volumes S around q will “stiffen” enough that we will see a simpler physic s there. We will not want our infinitesimal S to bec ome totally rigid, lest we never rec over any flexibility in the larger bodies B to whic h it belongs, but perhaps a small S might move like a little mechanism, so that some of the tec hniques of the previous sec tion bec ome applic able. For example, a standard weighted beam c an be assigned a small-sc ale mec hanic al element that eventuates in the stoc k Bernoulli-Euler equation for suc h struc tures (figure 2.31). Page 30 of 45

What is “Classical Mechanics” Anyway? Figure 2.31 Figure 2.32 In this situation, our element is allowed to turn about its c entroid, as well as move up and down in a plane, although a series of springs sets up an internal resistanc e to turning. In addition, gravity ac ts in the c enter of the element according to the weight W it bears. In this situation, Euler's rules for torque play a role in the derivations. Exc ept in early works,4 0 it is fairly rare to see presumed “mec hanic al elements” dec ked out in bloc ks and springs quite like this. But there are several alternate modes of presentation that c an ac hieve c omparable results by invoking the c ontrolled-virtual- work behaviors that we briefly disc ussed in the previous sec tion (figure 2.32). Thus, we might portray our Bernoulli-Euler element as illustrated, where we have an element that is intrinsic ally flexible, but whic h responds to c ontac t trac tions only at spec ific sites. As stated before, suc h restric tions represent a diagnosis of how applied thrusts are expec ted to transmit themselves through the element. It is evident that we get our required “simplific ation in the small” through loc ating these sites of c ontrolled thrust; otherwise, we would simply be looking at a small sec tion S of the original blob B we began with, displaying exac tly the same behavioral complexities as where we started. Modern books in engineering—at least, the sophistic ated ones—no longer follow these old polic ies, whic h trade upon rigid-body mechanics as an intermediary. From a prac tic al point of view, suc h presentations leave us rather c onfused as to whic h behaviors are possible—and whic h are not possible—within a c ontinuous material. A c ommon Page 31 of 45

What is “Classical Mechanics” Anyway? model for a drumhead, in effec t, c onstrains its movements in the mode of the bloc ks-and-c ords c onstruc tion illustrated in figure 2.33. Figure 2.33 Figure 2.34 Its little elements have been linked together in suc h a way that they c an only move up and down, but not horiz ontally. Translating those limitations into wave terms, this means that suc h membranes c an transmit only transverse waves (like surface water waves) but not compression waves (like sound). But are such materials really possible, exc ept in c oarse approximation? This is the kind of inductive guidance with respec t to the behavioral c apabilities of materials that we would like c ontinuum mec hanic s to provide. The strangeness of our drumhead's hypothetic al c apac ities c an be made quite vivid if we c onsider its one- dimensional analog, an oddity that lies c onc ealed within the basic equation for a vibrating string disc ussed in every c lassic al physic s primer (figure 2.34). In its derivation we tac itly posit that, in its stretc hing eac h sec tion of string “remembers” its rest position well enough to remain c onstantly above it, never veering left or right in the manner of the gray arrow. How c an a dumb piec e of string ac hieve this remarkable feat? In the drumhead c ase, we surreptitiously employed the rigidity of the bloc ks to enforc e the vertic al-only movements, inserting c ords to allow eac h element to bec ome effec tively longer as it does so. But our string lac ks any c omparable enforc ement mec hanism of this kind. Should we c onc lude that no c ontinuous material c an truly behave as our textbook model presc ribes or simply that it is unlikely, exc ept in c rude approximation? In fac t, nearly all of the standard c ontinuum models studied in undergraduate primers c ontain some hidden dimension of unlikely behavior of this ilk: they continually ask beams to bend in a plane, say, but to not bulge outward as they do so. But see if you can find a real material that will be so obliging. On the other hand, real materials do display odd abilities to “remember” their earlier states. If we attempt to find an infinitesimal mec hanism-like element that duplic ates these c apac ities, we are likely to require strange, Rube Goldberg-like devices. So, at base, our “problem of the physic al infinitesimal” is one of delineating, with some measure of c onfidenc e, the full range of infinitesimal behaviors that c an be legitimately expec ted of the points q within a c ontinuous body. The great twentieth-c entury investigations into the foundations of c ontinuum mec hanic s led by Clifford Truesdell and his Page 32 of 45

What is “Classical Mechanics” Anyway? sc hool dec ided that traditional approac hes of the c harac ter we have surveyed had jumbled together three basic tasks that should be kept distinc t: 4 1 (A) to establish the loc al existenc e of stress, strain, rate of deformation, and allied tensors within a c ontinuous body; (B) to supply “c onstitutive relationships” that c apture why a material like iron differs so greatly from putty or water; and (C) to exploit empiric al determinations of the dominant patterns of thrust propagation within a medium to render the results of tasks A and B more mathematic ally trac table. Ac c ording to this modern reassessment, the polic ies pursued by the great nineteenth-c entury masters of c ontinuum physic s (Kelvin, Stokes, and others) had mixed approximative considerations properly reserved for task C together with the general theoretic al princ iples required for Tasks A and B. Suc h blurring made it impossible to answer our “what range of infinitesimal behaviors are possible?” question with any c onfidenc e. Figure 2.35 To get a better sense of what is at issue here, let us return to our old problem of how to c ombine the traction vectors ac ting upon a surrounding shell ∂S modelings with the body forc es (inc luding ac c elerations) that ac t inside S. In partic ular, let us c arve out a finite internal volume S of a body B with an imaginary Eulerian c ut. As before, sum (= integrate) all of these ac tors as resultant forc es F* and torques τ* over S or ∂S ac c ording to need, just as we did with rigid bodies. But where inside S do F* and torques τ* ac t? What representative point should be appropriate for the finite volume S? In the c ase of rigid bodies, the answer did not matter bec ause of the rigidity, but lumps of putty will ac t quite differently ac c ording to where F* is plac ed. Onc e we establish how S as a whole behaves, we might be able to assign it some reasonable representative c enters (its c enter of gravity, perhaps) but, right now, suc h c enters move around inside S c onsiderably ac c ording to how the blob is affec ted by the outside forc es. It is at this stage that traditionalist approac hes invoke rigidific ation or little mec hanisms within S's that are suffic iently firm to allow our F* and τ* to work upon them in a more determinant manner. But to gain this firmness, the traditionalists invoke constraints and other modeling restric tions that our modernists regard as approximative and wish c onsigned to the “simplify the mathematic s” purposes c harac teristic of task C's portfolio. Ac c ordingly, the modern approac h advises us to overlook these “how do we halt the regress” c onc erns for the moment and assures us that we c an nonetheless regard Euler's two basic laws of motion (or balance principles, as they are usually c alled in this c ontext) as fully applic able to (almost) any Eulerian c ut S. This is a rather abstrac t c laim to ac c ept, due to the fac t that we possess little c onc rete sense of where or how F* and τ* will operate upon S. “Have patience,” our modernists advise, “we'll trap it eventually.” Crudely speaking, the proposal is that if we c ontinue shrinking S to ever smaller dimensions, in the final limit, we will rec over those infinitesimal c ubes C we considered earlier (figure 2.35). In fact, these C's are so small that they no longer qualify as Eulerian S's at all (the S's possess finite volumes, whereas our C's c omprise “dec orated points”). Due to their minute c harac ter, suc h C's will possess one trac tion vec tor on eac h fac e and only one (summed) body forc e vec tor and ac c eleration inside. Furthermore, the trac tions upon opposing fac es must be diametric ally opposed lest our C c ube find itself subjec t to an infinitesimal turning moment. The net effec t of these forc es is to make C either alter its volume or sheer, or some c ombination of the two. Onc e we know what happens here, then we c an determine what happens in c uts with larger volumes S by simply integrating all of the infinitesimals C's that c omprise it. Page 33 of 45

What is “Classical Mechanics” Anyway? Figure 2.36 These proc edures probably sound obsc ure (or even mystic al) due to the fac t that I have framed the proposal in the language of infinitesimals. So let us purge those notions from my presentation using tensorial objec ts instead (the basic tec hnique for doing so is rather abstrac t but beautiful). To do this we must understand how stress and strain tensors func tion. I will begin with the latter, c onventionally designated by ε. Take a point q inside a finite blob S and run an oriented referenc e plane through it (the orientation is supplied by the little gray arrow) (figure 2.36). Our strain tensor intuitively provides, in the guise of a matrix of nine numbers, how muc h the c orresponding fac e (or “response plane”) of an infinitesimal c ube at q has expanded or c ontrac ted (ac c ording to whether the c enter of the response plane has moved outward or inward from the referenc e plane) and also the degree to whic h the response plane has bec ome tilted with respec t to that original orientation (obviously it c an tilt in both x and y direc tions). In other words, a strain tensor is a giz mo that maps planes through points q to new planes (this is part of its proper definition). Employing this strain tensor information about the response planes through q, we c an, in effec t, rec onstruc t our original strained infinitesimal C by c alc ulating the dilation (= c ompression or expansion) and reorientation experienc ed by various c hoic es of referenc e frame as we run them through q. Now there needs to be a gradualist c oherenc e among our answers for we want our rec onstruc ted “infinitesimal” to turn out to be a skewed c ube and not, say, a skewed dodec ahedron. We enforc e this c oherenc e among our answers through the standard “vec tor spac e” qualities demanded of any tensor. The upshot of all of this is that the strain tensor attac hing to q c an be fairly c harac teriz ed as “the ghost of a vanishing shape”—the tec hnique c aptures the data that we need to have installed at q in a manner that explains why we are intuitively inc lined to pic ture q's strained state as an infinitesimal c ube with sides.4 2 Figure 2.37 We employ the same tec hniques to make sense of the stress tensor σ at q, exc ept that σ now plac es a tilted trac tion vec tor F upon our referenc e plane (figure 2.37). The c omponent of F that runs normal to the referenc e plane represents the pressure (c ompressive or dilatory) that strives to alter the volume of S; the planar c omponent of F c aptures its sheering c apac ities. Bec ause we normally do not want to deposit any unbalanc ed torques on S, we require the F on the other side of our c ube (= a referenc e plane with a reversed orientation) to be equal and opposite in magnitude. Operationally, this requires that the Page 34 of 45

What is “Classical Mechanics” Anyway? matrix of numbers c orresponding to σ must be symmetric , with only six independent values. In any c ase, our ε and σ tensors provide the basic information we require within our infinitesimal c ubes,4 3 while esc hewing any talk of infinitesimals per se. I hope it is evident that, while the tensorial method for esc hewing infinitesimals is quite c lever, most of the entangled diffic ulties within our physic al infinitesimal pac ket have been left untouc hed, for they largely c onc ern the question of the loc al traits that need to be deposited at q for c ontinuum mec hanic s to work c oherently. Onc e those physic al issues have been resolved, any “infinitesimal” proposal c an be easily reworked into a c ollec tion of tensors or allied objec ts. Figure 2.38 Using this language, the result of enforc ing Euler's two laws of motion upon (almost) every c ut S we c an c arve out of a body B tells us that stress and strain tensors will be loc ally defined at (most) points q inside B and will, furthermore, obey Cauc hy's c elebrated law of motion: Here the divergenc e operator (div) evaluates how the stress field varies in the vic inity of q and provides us with a vec torial assessment of where the greatest c hanges in σ lie.4 4 This provides us with a density vec tor that c an be meaningfully summed with the body forc e densities that ac t at q. Observe that Cauc hy's princ iple looks very muc h like Newton's sec ond law as it appeared within our point-mass setting and many authors identify it as suc h (although that c an be only regarded as a rather diffuse “family resemblanc e” c laim, in that we are plainly dealing with a c onsiderably more sophistic ated c onstruc tion now). Indeed, it is a mistake—although many elementary textbooks enc ourage the opposite point of view—to assimilate notions like stress too glibly to more straightforward notions like force (I devoted a fair amount of spac e to their proper mathematic al nature for this reason). Thus, many writers will assure their readers that stresses “reflec t the short range forc es within a material,” whic h is true in some loose “stresses reflec t information about suc h arrangements” sense (in a fashion that enc ourages us to c onc eptualiz e the underlying material in molec ular terms). But there is no ready rec ipe that c onverts these molec ular short-range forc es into the numeric al values that belong to the stresses assigned to points q within a c ontinuum modeling of the situation. Perhaps this last point c an be c larified with a spec ific example. The short-range forc es ac tive within most real materials rarely bind them into perfec t lattic es, but tolerate the irregularities known as disloc ations (figure 2.38). Large numbers of these lattic e defec ts c an emerge at sc ale lengths that need to be treated as short-range and c an affec t the mac rosc opic qualities of a material in signific ant ways. How, in a c lassic al c ontinuum modeling of our material, should its disloc ational properties be registered? A lot of rec ent work in extended c ontinuum mec hanic s (I will disc uss some of this later) has supplied a variety of answers to this question. In some of these sc hemes, the disloc ations are not c aptured in the material's strain tensor at all, but within other mathematic al c onstruc tions attributed to the point q (e.g., to a torsion within the underlying manifold on whic h q lives). Suc h a torsion c an be rec ogniz ed as the “short-range forc es” within the material just as ably as does its c onventional strain, but follows a different c oding sc heme. In truth, when we c asually parse stresses as short-range forc es, we are tac itly making a lift from c ontinuum Page 35 of 45

What is “Classical Mechanics” Anyway? mec hanic s into a different c onc eptual arena within whic h the tric ky notion of stress c an be “rationaliz ed” through a rough alignment with a more readily understandable form of material struc ture. Suc h lifts (whic h are a c ommon oc c urrenc e in c ontinuum mec hanic s) are fully in ac c ord with the theory-fac ade c harac ter of “c lassic al mec hanic s” overall, but they c an obsc ure the fac t that, c onsidered in their own terms, tensor fields are novel mathematic al c onstruc tions with their own spec trum of c harac teristic s (mathematic ians did not isolate the notion c learly until the end of the nineteenth c entury). Indeed, it is prec isely these spec ial qualities that allow modernists to halt our “labyrinth of the c ontinuum” regress in a novel way: they c laim that the trac tion vec tors around shrinking S's will deposit a loc aliz ed residue on q in the form of a stress tensor. With the help of a simple divergenc e c omputation, we c an then extrac t a vec tor to add to the body forc e and ac c eleration in a mathematically coherent manner. If we survey the c onc eptual framework just sketc hed, we realiz e that none of the fundamental princ iples employed direc tly c onc ern points q, but instead talk, in sometimes very abstrac t ways, about how finite volumes S behave. Thus Euler's two laws of motion hold only of finite “c uts” S extrac ted from a body B; they do not make sense for individual points q. Conservation of mass, likewise, c onc erns how finite blobs S relate to the referenc e manifold. And so on. Mathematic ally, suc h princ iples need to be expressed by integral differential equations, not as loc aliz ed differential equations per se. Cauc hy's law, to be sure, is of the latter c lass but it has been derived from fundamental integral princ iples; it has not been posited as basic . Within the point-mass setting of sec tion 3, partic ular materials were c redited with behavioral individualities through c hoosing the number of partic les present within the system and assigning them material c onstants (mass, c harge, etc .). These c onstants then turned on an appropriate set of spec ial forc e laws in modeling, suc h as Newton's law of universal gravitation. Modeling spec ific ations of this c harac ter we c alled “c onstitutive assumptions,” sec retly borrowing terminology from a c ontinuum c ontext. When we have suc c essfully assembled a c losed equational system by these proc edures, we say that we have thereby followed “Euler's rec ipe”. Although this portrait of modeling tec hniques within physic s is both simple and appealing, in point of brute fac t one regularly finds prac titioners evading the rec ipe's dic tates through appeal to ΔL* level c onstraints and allied modes of physic s avoidanc e. Indeed, those methodologic al intrusions have bec ome so pervasive in prac tic e that most point-mass modelers appear to forget that they have any obligations to trac k down a full set of spec ial forc e laws at all. But the appeals that typic ally displac e spec ial forc e laws within suc h c ontexts sc arc ely seem lawlike in their own right: “X is a rigid rod” does not sound muc h like a law of nature. In that sense at least, it is misleading to insist that suc h physic ists and engineers are seeking to find the laws to whic h nature c onforms. Within our c urrent c ontinuum-mec hanic s program, we are not allowed to invoke c onstraints in setting up our fundamental modeling equations. But what is the present analog to our former “spec ial forc e laws”? The stress/strain c onstitutive assumptions we have just examined. “But there are z illions of these,” we might protest. “Don't workers in c ontinuum mec hanic s attempt to reduc e their multitude to a smaller c ollec tion?” The answer is no, they don't; they merely try to sort the possibilities into general c lasses, so that the simplest forms of stress/strain behaviors c an be studied first. In other words, they supply a taxonomy of possible c onstitutive behaviors, but no reduc tive listing of spec ial forc e laws is ever offered. Indeed, in typic al modeling prac tic e, the c onstitutive princ iples assigned to a material are generally determined through direct experimentation on large hunks of the material in the laboratory. In this manner, in a Cauc hy-rec ipe modeling, its c ore c onstitutive equations reflec t a projection of behaviors witnessed experimentally at a large ΔL* length sc ale down to an infinitesimal scale. Page 36 of 45

What is “Classical Mechanics” Anyway? Figure 2.39 Let us finally turn to our Task C. Ac c ording to the modern program under review, we should not invoke c onstraints of any sort in setting up our basic c onstitutive modelings. But this methodologic al prohibition is c ommonly violated within traditionalist presentations of c ontinuum mec hanic s. Consider again the illustrated infinitesimal element for a Bernoulli-Euler beam (figure 2.39). Note that the applic able pushes and pulls upon the element are assumed to balance along fibers running ac ross the material. This assumption represents a c onstraint on permissible behavior, of the same general c harac ter as we examined in the previous sec tion. Ac c ording to Cauc hy's modeling rec ipe, we should properly supply c onstitutive equations of a Hooke's law ilk able to insure that stresses will be largely c onveyed ac ross the element in this fashion. Great—but see if you c an fill out a matrix of c oeffic ients that will do this. The sad truth is that this task is not at all easy—in fac t, we have c anvassed this same problem already, in the humble form of the vibrating string. The c onstraint c ritic al to the simple “derivations” found in most c ollege textbooks maintains that string elements forever hover infallibly above their original rest positions. But try to find a set of stress/strain relationships that c an simulate this behavior approximately within a three-dimensional material. This is again a daunting task.45 Plainly, adhering to the foundational c larity demanded within our modern approac h plac es ghastly burdens upon beginners in c ontinuum physic s, for the path to the one-dimensional wave equation bec omes strewn with knotty mathematic al thorns. Suc h c onsiderations provide a rationale for instituting a new, approximative division of c ontinuum mec hanic s that investigates how our stric t Task A/Task B modeling requirements c an be profitably c irc umvented through the wise exploitation of ΔL* level c onstraint information. Suc h work frames a third Task C for c ontinuum mec hanic s: develop mixed-level modeling tec hniques that relate to the stric t c onstitutive-modeling requirements of “foundational” c ontinuum mec hanic s in the same manner as the evasive tec hniques of analytic al mec hanic s relate to Euler's rec ipe.4 6 From this point of view, we c an antic ipate that the c harac teristic emphases of analytic al mec hanic s will make a strong reappearanc e within prac tic al c ontinuum mec hanic s, for the simple reason that the former prac tic es the approximative art of exploiting ΔL* sc ale c onstraints to isolate the pathways of dominating activity within a c omplex ΔL-level medium. So it is not surprising that the lore of old-fashioned c ontinuum mec hanic s appears riddled with innumerable lifts into rigid-body mec hanic s, for the demanding requirements of Cauc hy's rec ipe needed to be relaxed before the equations that support the great eighteenth- and nineteenth-c entury advanc es in wave motion, and so on, c ould emerge into c entral foc us. Continuum modeling c ould have never gotten on its feet historic ally without the temporary assistanc e of rigidified infinitesimals and “little mec hanisms.” If d'Alembert, the author of the first PDE for a vibrating string, had felt obliged to deal with matrix equations c ontaining 21 independent c onstants, c ontinuum tec hnique would have been abandoned as stillborn at birth. All of this merely undersc ores the lessons we have noted with respec t to the sec ret c ontribution of lifts with respec t to c lassic al mec hanic s’ triumphant hegemony. But the spec ific c onstraint-assisted lifts that helped traditional c ontinuum modelers on their way had the c urious effec t of enc ouraging themes within the philosophy of sc ienc e that c ontinue to reverberate strongly even to this Page 37 of 45

What is “Classical Mechanics” Anyway? day. They trac e to the following fac tors. In order to bloc k the “never simplifying” regress c reated by flexible materials that behave identic ally on all siz e sc ales, traditional modelers assumed that small portions of a material behave like little mec hanisms. In doing so, they tac itly c redited the lower sc ale lengths of a material with characteristics that they did not believe they really possess. It then appears that we cannot set up coherent “foundations” for flexible bodies without injec ting patent descriptive fictions to arrest an otherwise vic ious regress. And so the thesis emerges that physic s c annot begin its desc riptive tasks until it has first indulged in a preliminary degree of essential idealization: smallish portions of materials must be c redited with patently inc orrec t c harac teristic s. After we reac h a c ompleted modeling, we c an throw away the idealiz ed ladder we have c limbed, for our final equations will desc ribe materials that behave identic ally at every sc ale. But on route there, we must ac c ept, in the physic ist J. H. Poynting's phrase, a fic tive “sc affolding from without.”4 7 The assumption that some form of essential idealiz ation must be invoked to arrest c ontinuum physic s’ “labyrinth of the c ontinuum” problem has played an important, if often unac knowledged, role in shaping the doc trines of the philosophers who pondered the problems of c lassic al matter c arefully: Leibniz , Kant, Duhem, Hertz , Mac h, and others.4 8 Its enduring legac y is the lingering presumption that intentional misdescription represents a c ommonplac e ac tivity in sc ientific ac tivity. In retrospec t, however, this philosophic al thesis seems to have engendered by the lifts required to link Task A/Task B modeling demands with the more relaxed standards required in prac tic al work of a Task C c ast. Figure 2.40 Figure 2.41 In the naïve form we artic ulated, our Task A approac h to mec hanic s presumes that Euler's two laws hold true, in an abstrac t manner, for any c ontrac ting sequenc e of c uts S, S′, S, … surrounding a target point q. And this presumes that their respec tive perimeters ∂S, ∂S′, ∂S, … c an c arry full c omplements of trac tion vec tors. But this demand is too strong, partially bec ause some ∂S are too irregular to bear suc h measures, but also bec ause suc h requirements need to fail when ∂S cuts through a portion of shock wave surface (some irregularity must prevent the contracting c uts S, S′,S, … from installing stress and strain tensors upon these problematic points). In point of fac t, the c anonic al modelings of traditional mec hanic s have long tolerated funny spots, namely singularities, upon their boundaries. For example, take a notched rod and pull upon its two open faces with a uni- form tension (figure 2.40). The result is an infinite twist along the base of the c ut. Modern treatments of c ontinua employ rather fancy tools from functional analysis, such as trace operators, to bring the inner and boundary desc riptions of c ontinuous bodies into better mathematic al ac c ord. Very subtle c onsiderations with respec t to energy storage typic ally lie in the bac kground of suc h interior/boundary “harmoniz ations.” Page 38 of 45

What is “Classical Mechanics” Anyway? Figure 2.42 Here is an example of this phenomenon that I regard as partic ularly telling. Pull a knife through some water. The knife draws the top layer of the water with it (figure 2.41). Intuitively, we expec t that, after a c ertain period of mixing, the waters on the two sides of the c ut will soon fuse together. But ac c ording to the story that the PDEs of c ontinuum mec hanic s tell, this wound c an never heal, for differential equations c annot alter the topologies of the flows they trac k. But these desc riptive limitations entail that, without some signific ant alteration, the mathematic al framework of orthodox c ontinuum mec hanic s c an model neither the fusion nor the frac ture of ordinary materials (whic h is why the subjec t traditionally c onfines its attention to nonc omposite blobs in c irc umstanc es where they are unlikely to suffer frac ture or fission. To anyone who has not sc rutiniz ed the standard lifts of mec hanic al tradition in the c ritic al manner of this essay, this c laim will seem outrageous: “Of c ourse, c lassic al mec hanic s c an readily handle mixing: the molec ules from eac h side of the c ut rapidly intermingle until it bec omes impossible to determine where the dividing boundary had been” (figure 2.42). Yes, but observe that in this rationaliz ation we have esc aped into a reontologiz ed ΔL domain governed by point-mass mec hanic s or something similar. It is “c lassic al mec hanic s” all right, but it is not the same c ontinuum mec hanic s with whic h we started. Due to these readily available lifts, one c an learn a substantial amount of fluid mec hanic s without realiz ing that one's PDE tools are limited in this way.4 9 Although I have here disc ussed suc h issues in rather formal terms, many of the great historic al philosophers of matter (e.g., Loc ke, Leibniz , Kant) c ommented upon the fac t that the everyday proc esses of c ohesion and disassoc iation appear very mysterious from a mec hanic al point of view. Only the point-mass approac h handles suc h topic s with any satisfac tion. Yet it is unable to equip materials with the c harac teristic s they need when they are not about the business of breaking or fusing. The only route to a satisfac tory c overage of c ommon forms of everyday material behavior is to weld together a c lassic al mec hanic s from different desc riptive platforms assessable to one another along suitable esc ape-hatc h ladders. 6. Co nclusio n In sum, if we go searc hing for the “foundational c ore” of c lassic al physic s prac tic e in Hilbert's manner, we are likely to feel as if we have bec ome trapped in a novel by Kafka, with partic ular branc hes of a vast bureauc rac y c laiming greater authorities than they truly possess and, when c hallenged, shunting us off to other departments that assist us no further in our quest. And the most maddening aspec t of these unsettled c onvolutions is that the resulting interc onnec tions appear, when evaluated from the perspec tive of brute pragmatic s, as exc eptionally well plotted in their organiz ational arc hitec ture, for the intric ate interwebbing we c all “c lassic al mec hanic s” c omprises as effective a grouping of descriptive tools as man has yet assembled, at least for the purposes of managing the mac rosc opic aspec ts of the universe before us with well-tuned effic ienc y. In the final analysis, our investigations provide us with a ric her understanding of why “family resemblanc e” struc tures often possess great pragmatic utility. The c ruc ial point to observe is that the frequent lifts that populate the pages of c ollege textbooks do not func tion as the “derivations” their authors suppose them to be, but instead provide Task C-style guidelines for how diffic ult modeling problems c an be evaded through the exploitation of data (e.g., rigidity or princ ipal direc tions of thrust propagation) extrac ted from observation along a mixture of sc ale lengths.50 So while we have been c ritic al of suc h textbook lifts when evaluated from a Hilbertian point of view, these same passages perform a c ruc ial pedagogic al purpose in direc ting a modeler's efforts to loc ally effec tive results. In the final analysis, it is the astounding suc c ess of these well-tuned models with respec t to the Page 39 of 45

What is “Classical Mechanics” Anyway? mac rosc opic world that insure that “c lassic al physic s,” as an important intellec tual ac tivity, will probably remain with us forever. So while it is important to rec ogniz e, from a methodologic al point of view, that the routes whereby standard textbook prose stitc hes the fabric of “c lassic al mec hanic s” into a well-engineered fac ade rarely c omprise “derivations” in a proper sense, the good offic es they perform for us should not be devalued in rendering that judgment. I trust that many readers had the uneasy sense, when we c ritic iz ed worthy textbooks earlier for failing to satisfy Hilbertian standards of rigor, that somehow our target authors were “doing the right thing” in their presentations regardless. Yes, but suc h passages serve a different organizational purpose than we have been led to expect. Although we cannot properly explore the possibilities here, deeper answers are still wanted as to why the c harac teristic ingredients of “c lassic al mec hanic s” bind together into a fac ade as effec tively as they do. Although Wittgensteinians sometimes c laim otherwise, our remarkable c apac ities to sort human fac es into “family” groups wants explaining: the brain must perform some rough form of statistic al analysis over fac ial features when it c omputes its groupings, although the psyc hologic al mec hanisms involved do not appear to be well understood at present. Just so: the strong feelings of “family resemblanc e” with whic h every student of c lassic al physic s is familiar merit probing in the same vein. Tait invokes the phenomenon well: [A]ll who have even a slight ac quaintanc e with the subjec t know that the laws of motion, and the law of gravitation, c ontain absolutely all of Physic al Astronomy, in the sense in whic h that term is c ommonly employed: viz ., the investigation of the motions and mutual perturbations of a number of masses (usually treated as mere points, or at least as rigid bodies) forming any system whatever of sun, planets, and satellites. But, as soon as physic al sc ienc e points out that we must take ac c ount of the plastic ity and elastic ity of eac h mass of suc h a system, the amount of liquid on its surfac e, … [etc .], the simplic ity of the data of the mathematic al problem is gone; and physic al astronomy, exc ept in its grander outlines, bec omes as muc h c onfused as any other branc h of sc ienc e.51 Here Tait expresses his c onvic tion that point-mass physic s best enc apsulates the elusive “c entral c ore” to c lassic al mec hanic s, although he realiz es that this “c ore” must be dressed within the c onfusing garments of flexible bodies before reliable empiric al results c an be obtained. But what is the true nature of this “c entral c ore”? I believe that any reasonable answer must c ome from a deeper understanding of how our c lassic al desc riptive tools sit on top of quantum mechanics: the ways in whic h we usefully trac k mac rosc opic “work” and “energy” at the ΔL* level must somehow trac e to the ΔL-importanc e of c orrespondent notions within the quantum domain. However suc h issues resolve themselves, c lassic al mec hanic s, as studied here, offers many valuable lessons to philosophy as a whole: in partic ular, that well-wrought c onc eptual struc tures c an be assembled as fac ades tied together through “look ac ross siz e sc ales” linkages. But to praise a family-resemblanc e fabric in this manner is not to deny that its organiz ational patterns c an be ac c orded rational underpinnings. On the c ontrary, we should sc rutiniz e lifts and esc ape hatc hes within a fac ade with formal c are so that their operative strategies of physic s avoidanc e bec ome ac c urately identified and their empiric al outreac h ac c ordingly improved. As a prerequisite to those diagnostic endeavors, we must first rec ogniz e that the “derivations” provided in elementary textbooks rarely satisfy Hilbertian demands on rigor but instead fulfill the “look ac ross sc ale siz es” offic es that allow the basic terminology of c lassic al mec hanic s to c over wide swatc hes of mac rosc opic experienc e with an admirable effic ienc y. In these respec ts, c urrent work in c ontinuum mec hanic s provides an exc ellent paragon of how a useful base sc heme c an be profitably extended to wider applic ations onc e its c onc eptual supports have bec ome viewed without methodologic al illusion.52 Notes: (1) This is an extrac t (skillfully edited by Julia Bursten) of a longer survey to appear in a c ollec tion of essays entitled Physics Avoidance. I would like to thank Julia Bursten and Bob Batterman for their helpful advic e. (2) In textbooks, ontologically mixed circumstances (a point mass sliding upon a rigid plane) often appear. Usually these need to be viewed as degenerations of dimensionally c onsistent sc hemes (i.e., a ball sliding on a plane or a free mass floating above a lattic e of strongly attrac ting masses). (3) If a mathematic al treatment happens to make two point masses c oinc ide, that oc c urrenc e is generally viewed Page 40 of 45

What is “Classical Mechanics” Anyway? as a blowup (= breakdown of the formalism) rather than a true c ontac t. It is often possible to push one's treatment through suc h blowups through appeal to sundry c onservation laws and the rationale for these popular proc edures will be sc rutiniz ed in sec tion 3. (4) Modern investigations have shown that true ODEs and PDEs are usually the resultants of foundational princ iples that require more sophistic ated mathematic al c onstruc tions for their proper expression (integro-differential equations; variational princ iples, weak solutions, etc .). We shall briefly survey some of the reasons for these c omplic ations when we disc uss c ontinua in sec tion 4 (although suc h c onc erns c an even affec t point-mass mec hanic s as well). For the most part, the simple rule “ODEs = point masses or rigid bodies; PDEs = c ontinua” remains a valuable guide to basic mathematic al c harac ter. (5) Often internal variables suc h as spin are tolerated in these ODEs, even though they lac k c lear c ounterparts within true c lassic al tradition. (6) The abstract ruminations of The Critique of Pure Reason, for example, appear to have derived in part from the nitty-gritty worries about flexible matter that we shall review later. We look forward to Mic hael Friedman's big book on these issues. (7) Charles Darwin, The Descent of Man and Selection in Relation to Sex, Part II (New York: Americ an Dome, 1902), 780. (8) David Hilbert, “Mathematic al Problems,” in Mathematical Developments Arising from Hilbert Problems, ed. Felix Browder (Providenc e, RI: Americ an Mathematic al Soc iety, 1976), 14. (9) Georg Hamel, Theoretische Mechanik: Eine einheitliche Einführung in die gesamte Mechanik (Berlin: Springer Verlag, 1949). (10) Isaac Newton, Principia, vol. 1 (Berkeley: University of California Press, 1966), 349. (11) Hilbert, “Mathematic al Problems,” 15. (12) I do not have the spac e to survey suc h modern studies here, whic h attempt to, for example, rec over the tenets of rigid body mec hanic s from c ontinuum princ iples by allowing c ertain material parameters to bec ome infinitely stiff (thus “degeneration”). Generally the results are quite c omplex, with c orrec tive modeling fac tors emerging in the manner of Prantdl's boundary layer equations. Sometimes efforts are made to weld our different foundational approac hes into unity through employing tools like Stieltjes-Lesbeque integration. More generally, a “homogeniz ation” rec ipe smears out the detailed proc esses oc c urring ac ross a wide region ΔW in an “averaging” kind of way, whereas “degeneration” instead c onc entrates the proc esses within ΔW onto a spatially singular support like a surfac e (the Riemann-Hugoniot approac h to shoc k waves provides a c lassic exemplar). (13) After a suffic ient range of mec hanic al c onsiderations has been surveyed in later sec tions, we shall be able to sketc h a more favorable view of the useful offic es that standard textbook lifts provide. I should also add that we shall generally consider our “ΔL to ΔL* lifts” in two simultaneous modes: (1) as a modeling shift from one finite scale length to another (e.g., from ΔLG to ΔLO in our steel bar example) and (2) as a mathematic al shift from a lower dimensional object (a point mass or line) to a higher dimensional giz mo suc h as a three-dimensional blob. Properly speaking, these represent distinc t projec ts, although, in historic al and applic ational prac tic e, they blur together. (14) Stric tly speaking, a lift to c ontinuous variables from an ODE-style treatment involving a large number of disc rete variables at the ΔL level should not be c alled a “reduc ed variable” treatment, as we ac tually increased the number of degrees of freedom under the lift (normally, a true “reduc ed variable” treatment will supply a ΔL* level manifold lying near to some submanifold c ontained within the ΔL phase spac e). However, the desc riptive advantages of a lift to c ontinuous variables often resembles those supplied within a true “reduc ed variable” treatment, so in the sequel I will often c onsider both forms of lift under a c ommon heading. (15) In many statistic al problems, the population under review is artific ially inc reased to an infinite siz e, simply so that the applic able mathematic s will supply c risp answers to the questions we c ommonly ask. Left to its own devic es, mathematics is rather stupid in a literal-minded kind of way and finds it very diffic ult to answer questions Page 41 of 45

What is “Classical Mechanics” Anyway? in a “well, almost all of the time” vein, whic h is often the best that c an be ac hieved with respec t to a finite population. But if the same c ommunity is modeled as infinite, we c an often fool the mathematic s into supplying us with the brisk replies we desire. (16) Isaac Newton, op. cit. 13–14. (17) William Thomson (Lord Kelvin) and P. G. Tait, Treatise on Natural Philosophy, Vol. 1, (retitled as Principles of Mechanics and Dynamics) (New York: Dover, 1962), 219. (18) Newton, Principia Mathematica, 416. (19) In many c irc umstanc es, it is natural to borrow Coulomb's law from Maxwellian elec trodynamic s, but, stric tly speaking, this rule only suits static c irc umstanc es. Ac c ommodating dynamic c irc umstanc es within a “c lassic al physic s” frame, we must normally introduc e a foreign element (the elec tromagnetic field) that c arries us beyond the limits of our point-mass framework. Indeed, no one has yet figured out a wholly satisfac tory way to amalgamate c lassic al point masses with suc h a field. (20) Sometimes this phrase is tac itly restric ted by further requirements on the loc ations q(i,t): it seems strange to say that we have supplied a “c onstitutive modeling” for a c uc koo c loc k if we are willing to c onsider that “modeling” in a c ondition where its c omponent masses are sc attered ac ross the wide universe! (21) Z. Xia, “The Existenc e of Non-c ollision Singularities in Newtonian Systems,” Annals of Mathematics 135 (1992), 411–468. (22) For a more detailed disc ussion of these issues, see Mark Wilson, “Determinism: The Mystery of the Missing Physic s,” British Journal for the Philosophy of Science (2009), 173– 193. I might add that the c ommon tec hnique of dropping dimensions (e.g., c onfining point masses to a plane with no spec ific ation of the forc es that keep them there) should be c onsidered as a further variety of “Euler's rec ipe avoiding” polic y (suc h moves should be sc rutiniz ed with a c lose methodologic al eye whenever they are invoked). (23) Thomson and Tait, Treatise on Natural Philosophy, 1: 248. (24) Cf. the entries “Constitution of Bodies,” “Atom,” and “Attrac tion” in J. C. Maxwell, Collected Scientific Papers, ed. Ivan Niven (New York: Dover, 1952). Maxwell also worried that point-mass swarms c ould not remain struc turally stable when vigorously shaken or explain the fac t that the world's wide variety of materials only displays a very limited palette of spec tra. Reint de Boer, Theory of Porous Media (New York: Springer, 2000) provides a good capsule summary of these developments. (25) J. S. Rowlinson, Cohesion (Cambridge: Cambridge University Press, 2002), 110– 126 (this book is an exc ellent introduc tion to the fasc inating c onc eptual problems that attac h to “c ohesion” generally). (26) If no kinetic energy is lost to heat (a so-c alled “purely elastic c ollision”), then we possess enough “c onservation laws” (energy and linear momentum) to guide two c olliding point masses uniquely through a c ollision (as every elementary c ollege text demonstrates). But these princ iples alone are not adequate to three-way c ollisions, energetic losses, or to more oblique modes of sc attering. (27) Constraint relationships are sometimes maintained through fac tors external to the devic e (suc h as the pressures of an ambient fluid or the gravitational attrac tion that binds a c am to its follower), in whic h c ase the devic e is said to be force closed. Desc artes, for example, essentially dissec ted the universe into c omponent mec hanisms, but they were usually held together through forc e c losure rather than internal pinning. (28) Suc h c ontac ts are further c lassified as “higher or lower pairs” ac c ording the c ontac ting geometry they implement. (29) Thomson and Tait, Treatise on Natural Philosophy, Vol. 2, §441, 2. (30) Indeed, it is not evident to whic h body the c ontac t point “belongs” (one needs to beware of making simplistic assumptions about “how points belong to bodies” in suc h c irc umstanc es). Page 42 of 45

What is “Classical Mechanics” Anyway? (31) It is c ommon to designate the external c losure of a body B with the notation “∂B.” (32) In this c ontext, “Euler's First Law” is often viewed as simply “Newton's Sec ond Law” in applic ation to rigid bodies. Credit for regarding the “F = ma” sc heme as a framework upon whic h “rec ipes” for differential equations for both forms of mec hanic s c an be built is historic ally due to Euler, not Newton. As we shall see, the analogous rec ipe for c ontinua relies upon a formula traditionally c alled “Cauc hy's Law,” whic h many writers regard as yet “another version of F = ma” (although it ac tually employs the tric ky notion of stress that Cauc hy originated). The similarities of these three “rec ipe” formulas support the strong “family resemblanc e” c harac ter of “c lassic al mec hanic s.” Terminologic al issues bec ome more c onfusing within the c ontext of c ontinua, in whic h analogs of Euler's two laws are also applied to the sub-bodies in the interior of c ontainer blobs. In suc h c ontexts, these analogs are often dubbed the “balanc e princ iples” for momentum and angular momentum. In the c ontext of rigid bodies, onc e spec ific values for moments of inertia et al. have been c omputed with respec t to suc h entities, these values remain the same, allowing the import of Euler's princ iples to be expressed as equations of ODE type. Within flexible bodies, in c ontrast suc h values fluc tuate as they flex and so PDEs are required to c apture the requisite relationships. (33) Although I have quoted Lagrange's princ iple in its standard textbook form, it c onc eals a subtle ambiguity; spec ific ally as to whether the “r” c ited is a true position c oordinate or rather represents something “generaliz ed” like an angle. If the latter (whic h is usually what is needed), then the c orresponding “mass” terms “m” must be read as moments of inertia, etc . Presumably, we require some instruc tion in how these “generaliz ed inertial terms” are to be found. Suc h unnotic ed shifts are often sites of signific ant “lifts” (and sometimes outright errors, whic h are c ommon in this branc h of mec hanic s). The restric tion to “virtual variations” is nec essary bec ause the mechanical advantages of most mec hanisms c ontinuously adjust as they move through their c yc les. This means that inputted forc es F1, F2 , F3 on our c rane will not be able to balanc e quite the same output forc e F4 when the mac hine stands in a different c onfiguration. But the “instantaneous work” performed by the input forc es will always equal the “instantaneous work” expended at the outputs, whic h is the key idea that we need to c apture in our “virtual work” formula for static situations. (34) Donald T. Greenwood, Classical Dynamics (New York: Dover, 1997), 16–18. I do not intend these remarks to be as c ritic al as they may presently seem. Eventually, we c ome to see Greenwood's “proofs” as func tioning, not as derivations proper, but as “Task C” indic ators of profitable ways to avoid ΔL c onstitutive assumptions through the exploitation of knowledge of a material's ΔL* behaviors (spec ific ally, its apparent “rigidities”). (35) Ibid., p. 16. (36) Cornelius Lanc z os, The Variational Principles of Mechanics (New York: Dover, 1986), 70. (37) Cf. Clifford Truesdell, “The Creation and Unfolding of the Conc ept of Stress,” in Essays in the History of Mechanics (Berlin: Springer-Verlag, 1968: 184– 238). One needs to be wary of framing one's c onc eption of these notions from one-dimensional c ontinua suc h as strings or lamina, for in suc h reduc ed c ontexts “stress” does appear like a simple forc e density. In the main text, I am trying to bring forth the funny kind of three-dimensional structuring that is inherent in the notion of a “tensor.” (38) This proviso is enforc ed within a PDE modeling through the Saint-Venant c ompatibility equations. (39) One can witness some of this struggle in Kant's Metaphysical Foundations of Natural Science (Cambridge: Cambridge University Press, 2004) where he is plainly aware that some sourc e of sheer is needed to make sense of c onventional “solidity,” but c annot find a way to inc orporate suc h a quantity into his desc riptive framework. (40) I have patterned my first Bernoulli-Euler “element” after a diagram that Leibniz provides for a loaded beam. Cf. Clifford Truesdell, The Rational Mechanics of Flexible or Elastic Bodies 1638– 1788, (editor's introduc tion to Euler, Opera Omnia II, vol. 12) (Lausanne: 1954). (41) Clifford Truesdell, A First Course in Rational Continuum Mechanics (San Diego, CA: Ac ademic Press, 1991) and C. Truesdell and R. A. Toupin, “Classic al Continuum Physic s,” in S. Flugge, ed., Handbook of Physics, Vol. 3/i (Berlin: Springer, 1960:226–376). Morton Gurtin, Eliot Fried, and Lallit Anand, The Mechanics and Thermodynamics of Continua (Cambridge: Cambridge University Press, 2010) is also rec ommended and up to date. I should also Page 43 of 45

What is “Classical Mechanics” Anyway? indic ate that many researc hers outside of Truesdell's sc hool c ontributed to the new understandings we shall outline, without fully embracing the “purism” characteristic of the latter's approach. (42) Philosophers new to the peculiar world of continuum physics parlance should prepare themselves for phraseology suc h as “dimensionless point c ube” (J. D. Reddy, An Introduction to Continuum Mechanics (Cambridge: Cambridge University Press, 2008), 126—an excellent book, by the way). (43) A. N. Whitehead did some foundational work in mec hanic s at the turn of the twentieth c entury and his “method of extensive abstrac tion” was later populariz ed by Bertrand Russell in Our Knowledge of the External World (London: Routledge, 2009). I am not sure how Whitehead understood his construction (which shrinks in on points through dec reasing volumes), but Russell plainly regarded the tec hnique entirely as a logic al proc edure for “defining away points.” Russell's misunderstanding of the underlying physical problematic continues to reverberate within the halls of analytic philosophy. For a survey, see Mark Wilson, “Beware of the Blob,” in Dean Zimmerman, ed., Oxford Studies in Metaphysics (Oxford: Oxford University Press, 2008). A subtle point : when we combine our stress and strain information, should our resultant vectors situate themselves on the reference or the response planes? This matter becomes important in nonlinear elasticity and requires the c areful delineation of different stress tensors (“Piola-Kirc hhoff” versus “Cauc hy”) that one finds in modern textbooks. (44) ρg, it will be rec alled, c aptures the summed body forc es ac ting upon q. In following this standard representation, we are tac itly ignoring the third law demands that persuaded us to distinguish V(q) from V*(qn ) earlier (the mathematic s of c ontinua is rough enough without fussing about that!). It is important to realiz e that the ac c elerative term behaves mathematic ally very muc h like g and is often c alled an “inertial forc e” as a result (some of the third law ambiguities surveyed earlier trac e to this drift in the signific anc e of “forc e”). And an important symmetry with respect to constitutive equations is relevant as well: materials (usually) respond to an applied sc hedule of ac c elerations by exac tly the same rules as they reac t to a c omparable array of genuine forc es (this requirement is c alled “material frame indifferenc e” or “objec tivity”). (45) For a vivid illustration of the divergenc e between traditional methods and the approved “modern” approac h, see Stuart S. Antman, “The Equations for the Large Vibration of Strings,” American Mathematical Monthly 87 (1980). Drops in dimension through appeal to symmetries usually ac t in the manner of c onstraints. (46) As we have seen, traditional modelers commonly appealed to little mechanisms as a means of introducing Task C simplific ations into their modelings, so that analytic al mec hanic s serves as a c onvenient house of refuge for c ontinuum mec hanic s as well. (47) J. H. Poynting, “1899 Presidential Address to the Mathematic al and Physic al Sec tion of the British Assoc iation (Dover),” British Association Report (1899), 615–624. (48) As a c ase in point, a key doc ument within the rise of “anti-realism” is Karl Pearson's onc e influential The Grammar of Science (London: Thoemmes Continuum, 1992), which is very explicit in its continuum mechanics roots, c ommingled with a variety of neo-Kantian themes. (49) The disc ussion in Ric hard E. Meyer, An Introduction to Mathematical Fluid Dynamics (New York: Dover, 2007) brought home the point to me. (50) In the applic ations c onsidered here, only two c harac teristic sc ale lengths are generally relevant, but Batterman's essay in this volume surveys some of the exciting recent work that promises a capacity to intermingle data extrac ted from a wider array of sc ale siz es. (51) P. G. Tait, Heat (London: MacMillan, 1895), 9–10. (52) The c laim that everyday c lassific atory words operate along organiz ational princ iples similar to those surveyed here c omprises the c hief argumentative burden of my Wandering Significance: An Essay on Conceptual Behavior (Oxford: Oxford University Press, 2006). Page 44 of 45

What is “Classical Mechanics” Anyway? Mark Wilson Mark Wilson is Professor of Philosophy at the University of Pittsburgh, a Fellow of the Center for Philosophy of Science, and a Fellow at the Am erican Academ y of Arts and Sciences. His m ain research investigates the m anner in which physical and m athem atical concerns becom e entangled with issues characteristic of m etaphysics and philosophy of language. He is the author of Wandering significance: An essay on conceptual behavior (Oxford, 2006). He is currently writing a book on explanatory structure. He is also interested in the historical dim ensions of this interchange; in this vein, he has written on Descartes, Frege, Duhem , and Wittgenstein. He also supervises the North Am erican Traditions Series for Rounder Records.

Causation in Classical Mechanics Sheldon R. Smith The Oxford Handbook of Philosophy of Physics Edited by Robert Batterman Abstract and Keywords This c hapter disc usses c ausation in c lassic al mec hanic s and addresses the skeptic al argument that c ausation is not a fundamental feature of the world whic h was initiated by Bertrand Russell. It disc usses Russell's skeptic ism and explains that c onsiderations of c ausality are often invoked by physic ists when evaluating equations, inc luding c andidates for fundamental equations of c lassic al physic s. The c hapter identifies areas where the princ iple of c ausality applies. These inc lude Green's func tions, radiation theory, equations of motion, and dispersion theory. K ey words: cau sati on , cl assi cal mech an i cs, B ertran d Ru ssel l , sk epti ci sm, ph y si ci sts, cl assi cal ph y si cs, G reen 's fu n cti on s, radi ati on th eory , equ ati on s of moti on , di spersi on th eory Before the nineteenth c entury, it was c ommon to think that muc h of our understanding of the physic al world was organiz ed around the c onc ept of c ause and general “c ausal princ iples.” Ac c ording to David Hume, “All reasonings c onc erning matters of fac t [roughly, non-tautologous truths] seem to be founded on the relation of Cause and Effect” (Hume 1748, 16). Later, for Immanuel Kant, the c ategory of c ause was one of the pure c ategories of the understanding whic h the mind uses to struc ture its experienc es and without whic h c omprehension of a c oherent world would be impossible. By the late nineteenth c entury, however, it bec ame c ommon among physic ists and like-minded philosophers to assert that, at bottom, the c onc ept cause was not partic ularly important for understanding the physic al world bec ause c areful study of the physic al world had revealed causation to be absent from it. The bac kdrop in philosophy for most c ontemporary disc ussions of suc h skeptic ism about c ausation and its importis Bertrand Russell, who c laimed that c onsiderations of c ausation play no role in theoriz ing in advanc ed sc ienc es, espec ially physic s.1 Although Russell's disc ussion is c omplex and involves a number of distinc t c onsiderations, two of the main points seem to be the following:2 1. It is impossible to unequivoc ally identify “c auses” and “effec ts” within the fundamental equations of physic s, but physic s gets along just fine in spite of that fac t.3 This shows that c ausation is not a fundamental feature of the world. 2. There are no general c ausal princ iples that plac e restric tions on physic al behavior that are not otherwise there.4 With regard to the latter point, Russell c laims, “The law of c ausality, I believe, like muc h that passes muster among philosophers, is a relic of a bygone age, surviving, like the monarc hy, only bec ause it is erroneously assumed to do no harm” (Russell 1981, 132). Russell's explic it target—the “law of c ausality”—seems to be “same c ause, same effec t,” but the general tenor of his disc ussion suggests that he would also rejec t maxims like “everything has a c ause” and “the c ause c omes before the effec t.” These days, many think that Russell's skeptic ism is misguided.5 For, it is noted that physic ists frequently invoke Page 1 of 24

Causation in Classical Mechanics c onsiderations of “c ausality” when evaluating equations, inc luding c andidates for fundamental equations of c lassic al physic s. Although “c ausality” c an mean different things (e.g., determinism and restric tions on the veloc ity of c ausal propagation),6 I shall foc us in this essay on the maxim,“the c ause prec edes the effec t,” and on some plac es within c lassic al physic s where appeal to it allegedly enters.7 (My disc ussion will be limited to issues that arise already in a flat spac etime without c losed c ausal c urves.8 ) Here is a brief list of the plac es where it is c laimed that a “Princ iple of Causality” applies: 9 1. Advanc ed Green's func tions—to be desc ribed later—are often dec lared by physic ists to be unphysic al bec ause they are “ac ausal.” They suggest that the effec t c omes before the c ause. 2. In radiation theory, the Sommerfeld radiation c ondition is used to rule out waves c ollapsing in on a sourc e from infinity. Intuitively, these are waves that are c aused by the sourc e but are c aused “into the past.” So, we invoke the Sommerfeld radiation c ondition so as to adhere to c ausality. 3. Some equations of motion—for example, the Abraham-Lorentz Equation and the related Lorentz -Dirac Equation—are dismissed by physic ists bec ause they violate c ausality. 4. In dispersion theory, dispersion relations are shown to follow for systems whic h are c ausal. Sinc e one invokes c ausality early on in the theory of dispersion, it might be thought that c ausality is an important, fundamental princ iple in physic s. I shall disc uss eac h of these items more or less in turn with an eye toward examining what role—if any— assumptions of c ausality play. A c omplete weighing of the role of c ausality is out of the question in an essay of this scope. But, I hope to give at least a sense of the sorts of arguments that have been presented. 1. Identifying Causes and Effects: Advanced and Retarded Green's Functio ns Although Russell's skeptic ism is not obviously c oming from these quarters, one major driving forc e for c ausal skeptic ism of the sort that he advoc ates is the denial that there is any reason to privilege the so-c alled “retarded” Green's func tion for a system over the “advanc ed” Green's func tion. Green's func tions will be desc ribed in detail below, but roughly a Green's func tion desc ribes the effec t of an instantaneous impulse ac ting on a system. The retarded Green's func tion desc ribes the effec t as c oming after the c ause, whereas the advanc ed Green's func tion desc ribes the effec t as prec eding the c ause. As a soc iologic al matter, it is c ertainly not unc ommon for physic ists to announc e the privilege of the retarded Green's function or related notions like the retarded potentials. The following quotations will give some sense of the ubiquity of suc h c laims: • “Although the advanc ed potentials are entirely c onsistent with Maxwell's equations, they violate the most sac red tenet in all of physic s: the princ iple of c ausality.” (Griffiths 1981, 399)10 • “In the usual theory one omits the advanc ed solutions [or advanc ed Green's func tions] as [their] existenc e would be in violation of our ordinary c onc ept of c ausality.” (Weigel 1986, 194) • “The advanc ed potentials are mathematic ally allowed solutions, but they c onflic t with the basic physic al concept of causality.” (Johnson 1965, 21) • “For the time being, at least, we disc ard the advanc ed solution as unphysic al.” (Vanderline 2004, 272) I shall begin with the very basic example of an undamped harmonic osc illator, sinc e muc h of the problematic already arises in this simple c ontext. An example of a harmonic osc illator would be a bloc k attac hed to a spring having a linear restoring forc e with no damping present. The equation for a simple harmonic osc illator is just a c onc rete instanc e of Newton's Sec ond Law, F = ma: (1) where dots represent time derivatives and ω 0 is taken to be c onstant. In (1),f(t) is an “inhomogeneous term,” a term that does not involve the dependent variable (in this c ase x). Suc h terms are typic ally thought to represent a cause acting on the system from outside. There are several reasons for this thought: insofar as the inhomogeneous term does not involve the dependent variable, it is not representing something that is a func tion of the internal state of the system being modeled. But sinc e it makes a differenc e to the evolution of the system, it represents an outside influenc e on it. Another c onsideration is that a nonz ero inhomogeneous term will inc rease Page 2 of 24

Causation in Classical Mechanics (or dec rease as the c ase may be) the energy of the osc illator, but there should not be any suc h energy c hange in a system that is “c losed” (in the sense of being noninterac ting with its environment). Perhaps, we have found a rationale for thinking of f(t) as a c ause of c hanges to the system. Insofar as this is so, the first c laim frequently made by c ausal skeptic s— that one c annot identify c auses in physic s—appears to be overblown. However, a c onsequenc e sometimes drawn from this skeptic al c laim might still be thought to be c orrec t: Nothing terribly important hinges upon the identific ation of f(t) as a c ause for the purposes of doing mathematic al physic s; one c an solve the equation— given some f(t) and suitable c onditions leading to uniqueness —without thinking of f(t) as representing a c ause, and suc h a solution would give the entire evolutionary history of the osc illator. Sinc e one derives the entire trajec tory of the system without having to identify f(t) as a c ause of anything, it is not so obvious that the identific ation of c auses is partic ularly important for physic s. I shall, however, leave the question of what import this identific ation has to one side. For, even brac keting that issue, we still have not seen that we c an loc ate c auses and effects. What is the effec t of the c ause represented by the inhomogeneous term, f(t)? Sinc e suc h a term c an be rather c omplex depending upon what func tion f(t) is, it is easiest to think in terms of a “point-sourc e,” a sourc e that ac ts only at a single instant of time. This will be fruitful in this c ase, sinc e we have a linear equation, and one will be able to use this simpler inhomogeneous term to c ompose more c omplex ones. Suc h an instantaneous sourc e is represented by the Dirac delta func tion. A solution to (1) (along with other c onditions) when there is a Dirac delta func tion sourc e is known as a Green's func tion.11 Thus, the Green's func tion problem is based around the following equation, whic h tells us how a harmonic osc illator responds to a delta func tion kic k or gives the effec t of suc h a kic k: (2) Any g(t) that solves the Green's func tion problem is a Green's func tion. The easiest path to a Green's func tion is via Fourier transforms.12 The Fourier transform of a func tion, f(t), may be defined as follows: (3) Thinking for a moment of (1), we c an Fourier transform eac h term of the equation: (4) Using the fact that the Fourier transform of a derivative is − iω times the Fourier transform of the undifferentiated function, we get (5) where ˆx indicates the Fourier transform of x. Simple algebra gets us to (6) One hopes to arrive at the solution to equation (1) via inverse Fourier transform from (6): (7) Let us return to the Green's func tion problem where f(t) = δ(t). Up to a fac tor of 1 , the Fourier transform of δ(t) √2π has the value that eiwt takes at t = 0.13 By Euler's identity, this is just the value that c osωt + i sin ωt takes at t = 0, and this is just 1. So, for the Fourier transform of the Green's func tion, we derive (8) Page 3 of 24

Causation in Classical Mechanics Figure 3.1 Contours for harmonic oscillator Green's functions. To get “the” Green's func tion, one just needs to take the inverse Fourier transform: (9) The problem with the standard interpretation of this integral is that ĝ(ω) has poles on the real axis (at ω 0 =±ω). To make sense of the integral one may c hoose a c ontour that goes around the poles (like the c ontours in figure 3.1 where “x” represents a pole of ĝ(ω)). One then takes some limit so that the c ontour enc ompasses the entire real axis (exc ept for the dimples around the poles) with the c ontribution from the large semi-c irc le going to z ero. (More details about this will follow when the damped osc illator is disc ussed.) Depending upon what c ontour one takes, one gets either the retarded Green's func tion, gret, or the advanc ed Green's func tion, gadv, or some linear c ombination of them. Those func tions are as follows (Butkov 1968, 282): (10) (11) The first arises from taking the integral along a c ontour inc luding the real axis but with dimples that go over the poles (as in the left of figure 3.1). The sec ond arises from a c ontour with dimples that go under the poles (as in the right of figure 3.1).14 A linear c ombination of the two arises from taking a c ontour that goes over one pole and under the other. Insofar as these Green's func tions represent the response of the system to a Dirac delta func tion c ause, it seems reasonable to think that they represent the effec t of suc h a c ause for a harmonic osc illator system. Unfortunately, however, we have not found a unique effec t. We have, rather, two15 different evolutions assoc iated with the Dirac delta func tion c ause. The retarded Green's func tion suggests that the Dirac delta func tion kic k c auses harmonic osc illations after the kic k is applied to the system; the advanc ed Green's func tion suggests that similar osc illations are c aused before the applic ation of the kic k. A c ausal skeptic c an note here that we have failed to identify the effec t of the delta func tion kic k, but it is not c lear that we are the worse for it. The situation is analogous for Maxwell's equations desc ribing the behavior of the elec tromagnetic field in a vac uum. After some mathematic al work to dec ouple the equations, one gets the wave equation for the elec tric field (and an analogous equation for the magnetic field): (12) If one solves for “the” Green's func tion for the wave equation via suc h Fourier transform means, one finds that what one gets depends upon a c ontour sinc e, as in the c ase of the harmonic osc illator, the Fourier transform of the Green's func tion has poles on the real axis (Griffel 1981, 74; Barton 1989, 406). The retarded Green's func tion for the wave equation suggests that if the field is subjec t to a Dirac delta func tion kic k, a spheric al wave spreads out at the speed of light into the future. The advanc ed Green's func tion, on the other hand, suggests that suc h a wave c ollapses in on the sourc e at the speed of light. Unless one c an find some reason to privilege one of these representations of the effec t of a Dirac delta func tion kic k, we have failed to determine the effec t of suc h a kic k. 1.1 Privileging the Retarded Green's Function Equations (1) and (12) are insuffic ient to uniquely determine the effec t of the inhomogeneous term. Nevertheless, Page 4 of 24

Causation in Classical Mechanics there are two standard approac hes to selec ting the retarded Green's func tion as—in some sense—preferred. They apply similarly in the osc illator and wave equation c ases. The first is to think in terms of a different differential equation for whic h the Fourier transform proc edure sketc hed above gives a unique Green's func tion. The sec ond is to add additional c onstraints to the differential equation for the Green's func tion but without modifying the differential equation. Let me c onsider them in turn. 1.1.1 Adding Damping One standard “physic al motivation” for a partic ular c ontour of integration is that any real, mac rosc opic osc illator will have damping (Butkov 1968, 281). For example, for a bloc k attac hed to a spring, there would be internal damping in the spring but also from the ambient air. Sinc e suc h damping was not inc luded in the equation, there is missing physic s in the undamped osc illator equation. To remedy this, one turns to the damped osc illator equation. If we add dissipation (of a simple sort), we get (13) where γ 〉 0 and c onstant. As in the undamped c ase, we Fourier transform everything to get (14) Solving for the Fourier transform of x(t), we get (15) Thinking in terms of a delta func tion forc ing, we arrive at the Fourier transform of the Green's func tion for the c ase as before: (16) The Fourier transform of the Green's func tion for this c ase has poles in the lower half of the c omplex plane, not on the real axis.16 In fac t, adding damping, however small, to the undamped osc illator will bring the poles of the Fourier transform of the Green's func tion down into the lower part of the c omplex plane. Bec ause of this, one no longer has the issue of what c ontour to take around the poles lying on the real axis, sinc e the poles no longer lie on the real axis. Figure 3.2 Contours for the damped oscillator. From the inverse Fourier transform, one gets a Green's func tion for the damped osc illator that is retarded in the sense that it is z ero before the applic ation of the delta func tion kic k. To give a pic ture of how one arrives at this result, one takes the inverse Fourier transform of the Green's func tion (17) via c ontour integration. One starts by evaluating the integral over a c ontour from −R to R on the real axis along with a semi-circle either over or under that interval and then taking the limit as R goes to infinity. (See figure 3.2.) If one c loses the c ontour downward via a semi-c irc le in the lower half of the c omplex plane, it c an be made to include the poles of the integrand and, thus, by the Residue Theorem the integral will be −2πi times the sum of the residues of those poles (sinc e the c urve is oriented c loc kwise). One then shows that as R goes to infinity, the integral along the semi-c irc le will go to z ero when t 〉 0. So, in the limit, the integral over the c ontour is just the integral on the real axis, sinc e the c ontribution from the semi-c irc le goes to z ero, but the integral is nonz ero, sinc e there are poles within the c ontour. This will not, however, represent g(t) for t 〈 0 for, in that c ase, the integral over 17 Page 5 of 24

Causation in Classical Mechanics the c hosen semi-c irc le would not vanish; in fac t, it bec omes unbounded.17 However, if, instead, one c loses the c ontour of integration with a semi-c irc le in the upper half of the c omplex plane, one does get a representation of g(t) for t 〈 0 sinc e, in that c ase, the c ontribution of the semi-c irc le does vanish as one takes the limit as R goes to infinity. However, sinc e the integrand is analytic within that c ontour, one gets that the value of the integral is z ero. Thus, g (t) turns out to be 0 when t 〈 0. As suc h, one arrives via these means at a retarded Green's func tion. (For details of the derivation, see Butkov (1968, 277– 80); see also Wallac e (1984, 157– 160).) The motivation, then, for thinking that the retarded Green's func tion for the undamped osc illator truly reveals the c ausal direc tion of the system is the following: the equation for the undamped osc illator really is just an approximate model of a system for whic h the damping is quite small; in reality, there are no mac rosc opic , undamped osc illators. So, we know that the system we are modeling is ac tually damped. As suc h, the c orrec t Green's func tion is the retarded one, sinc e that is the only one that arises via this proc edure for the damped osc illator. So, the retarded Green's func tion c orrec tly represents the c ausal direc tionality of the system. We just started with an approximate equation, the one for the undamped osc illator. But, had we used a better equation, the one for the undamped osc illator, we would have gotten the c ausal direc tionality of the system right. Note that on this approac h the retarded Green's func tion for the osc illator is not privileged bec ause it is the only one that obeys the maxim, “the c ause is prior to the effec t.” Rather, it turns out that the c ause is prior to the effec t bec ause there is damping in any real mac rosc opic osc illator. So, it does not look as if there is any brute appeal to a “c ausality maxim” here. Rather, all of the work in privileging the retarded Green's func tion follows from the ac tual presenc e of damping. So, roughly, the c onstraint is: “If there is phenomenologic al damping, add damping to the model.” Onc e we invoke that, we do not have to invoke anything like “the c ause is prior to the effec t.” So, this is prima fac ie a c ase where one is not imposing “c ausality” but is imposing something else. Moreover, even if this were the right way to think of the privilege of the retarded Green's func tion in the osc illator c ase, it is not c lear that it is an appropriate way to think of all c ases of interest. In a vac uum, the c lassic al elec tromagnetic field is taken genuinely to be desc ribed by the undamped wave equation (12) even though the poles of the Fourier transform of the Green's func tion for it lie along the real axis. In this c ase too, if one adds damping, one ends up with a retarded Green's func tion. But, if we do not think that the undamped equation is just an approximation to a better equation involving damping, why would we think that we have learned something about the c ausal direc tionality of the elec tromagnetic field in a vac uum from c onsidering damped systems? One possible response is that we learn from other systems what the appropriate c ausal direc tionality is for this system bec ause from these other systems we learn about c ausal direc tionality tout court, c ausal direc tionality in nature as a whole rather than in this or that system. Sinc e waves in a material medium are damped, we learn that retarded Green's func tions properly represent c ausal direc tionality tout court. If it were the c ase that some materials were anti-damped and, thus, give rise to advanc ed Green's func tions via the Fourier transform proc edure, we would not be able to assign a unique c ausal direc tionality tout court.18 Rather, we would have to talk of the c ausal direc tionality of this or that system. But, if there are no anti-damped materials, we are able to assign a c ausal direc tionality tout court. As suc h, we are able to assign a c ausal direc tionality for waves in a vac uum, and that direc tionality is properly represented by the retarded Green's func tion. At this point, however, a c ausal skeptic will want to know what reason we have for thinking that there is c ausal direc tionality tout court. Why doesn't the fac t that we did not find unambiguous c ausal direc tionality in the wave equation show that there is not? The c laim that there is c ausal direc tionality tout court is motivated by the c laim that for all systems that have a privileged c ausal direc tion, it is the same direc tion. Even if that is true it does not change the fact that there are some systems that do not have one. On the other hand, there are some grounds for thinking that the equations that do not reveal a privileged c ausal direc tion should be our guide to whether there is c ausal direc tionality tout court. For, typic ally, the most fundamental equations of c lassic al physic s do not have the time asymmetry that appears in equations with damping. Thus, one might think that the undamped equations are a better guide to c ausal direc tionality in nature than the damped equations. However, they show the absenc e of c ausal direc tionality. As suc h, we still have not been able to uniquely identify effec ts. Of c ourse, we have found c ausal direc tionality in the damped osc illator equation. This suggests that it is not true that it is impossible to identify c auses and effec ts within some of the equations of physic s. But, if the equations that do not have c ausal direc tionality are more fundamental than the ones that do, one does not get the impression that Page 6 of 24

Causation in Classical Mechanics causal directionality is a fundamental feature of the world as is sometimes thought. 1.1.2 “Causality”: The Initial Value Problem There is another approac h to selec ting the retarded Green's func tion that is more c ommonly assoc iated with invoking a maxim of “c ausality.” Here, one does not look to a modified equation of motion, so no new physic s is added. Rather, one adds supplementary data that will selec t out the retarded Green's func tion as unique. In partic ular, one posits that the Green's func tion satisfies the following initial c ondition: For t 〈 0 (18) It is evident that this rules out the advanc ed Green's func tion and also any nontrivial linear c ombination of the advanc ed and retarded Green's func tions. Often, this c ondition is referred to as a c ausality c ondition. However, one might like a motivation for imposing this c ondition. A c ausal skeptic will not be too impressed with the c laim that we impose it so that the c ause prec edes the effec t, sinc e he is skeptic al about both the truth of that c laim and of its importanc e. As we shall see, a motivation for suc h initial c onditions c an be found, but onc e we see what it is we may note that it has little to do with c ausation per se and that c ertain elements of the buildup to it are optional. Let us start by thinking generally about the initial value problem. The general solution to an initial value problem for an inhomogeneous equation like (1) is the sum of the general solution to the assoc iated homogeneous equation— i.e., (1) with f(t) set to z ero—and a partic ular solution to (1). Thus, the general solution to (1) is as follows: (19) where xh(t) is the general solution to the homogeneous equation and xp(t) is a partic ular solution to the inhomogeneous equation. To get a unique solution, however, one typic ally assigns both the amplitude x(t) and the veloc ity x(t) at a partic ular time. Suppose one assigns suc h initial c onditions at a time t0 , before the inhomogeneous term turns on. In this c ase, a simple way to write the solution is to build the initial c onditions into the solution to the homogeneous equation in the sense that xh(t0) = x(t0) and ˙xh (t0 )  =  x˙(t0 ). Because it satisfies the initial c onditions, let us c all that part of the solution xi(t). The solution to the relevant inhomogeneous problem may now be written in terms of the Green's func tion as follows: (20) But, for this to work out, one needs some further c onditions upon g(t − t′) so that one does not end up ∞ contradicting one's initial conditions. What one does not want is for ∫ g(t − t′)f(t′)dt′ to have either a nonzero −∞ total value or a nonz ero veloc ity at the initial moment, t0. One c an intuitively see that if one were to add the advanc ed Green's func tion arising from a sourc e later than t0 to xi(t), one will have c ontradic ted one's initial c onditions. In more detail: How c ould the integral in (20) ac quire a nonz ero value at t0? In the setup, f(t) is z ero until after t0 only bec oming nonz ero at some later time t1; 19 let us suppose that it remains nonz ero until t2. The integral c an only ac quire a nonz ero value at t0 if g(t0 − t′) and f(t′) are nonz ero together (and are so on more than a set of Lebesgue measure z ero). If g(t0 − t′) is nonz ero on more than a set of Lebesgue measure z ero in the interval where t′ ranges from t1 to t2, then there risks being a c ontribution to the integral. However, all of the times in that interval are later than t0. So, g(t0 − t′) has a negative argument along that interval. If g(t − t′) takes only the value z ero for negative arguments, then the integral c annot have a nonz ero value at t0. So, one will want to think of the Green's func tion as satisfying the initial c ondition (18) in addition to (2). These c onditions leave only the retarded Green's func tion standing. Thus, the desired solution to the problem c an be written as follows: (21) Many view imposing c ondition (18) on the Green's func tion as imposing “c ausality.” However, the motivation for (18) given above has nothing to do with wanting to maintain the truth of any c ausal dic tum like “the c ause prec edes the effec t.” Instead, it had to do with not wanting to violate the princ iple of nonc ontradic tion: onc e we have assigned initial c onditions that represent the state of the osc illator at t0 and we have built those initial conditions into the solution to the homogeneous equation, the solution to the inhomogeneous equation that we add to it c annot be suc h as to c ontradic t them. It is a bit odd to c all this approac h the imposition of “c ausality” when the real driving forc e in the argument is the princ iple of nonc ontradic tion and not anything having to do with c ausation. Page 7 of 24

Causation in Classical Mechanics As a matter of fac t, quite often, physic ists are c onsiderably more lax about allowing advanc ed Green's func tions to play a role when there is no possibility of c ontradic ting spec ified state values. For example, advanc ed Green's func tions are often allowed to play a role in the derivation of an equation of motion for a c harged point-partic le that senses its own field—an equation that inc ludes a partic le's “self-interac tion” or “radiation reac tion.” Assuming that a c harged partic le with a nonz ero radius, a, gives rise to a purely retarded field as it moves in one dimension, the force on it from its own field is (Feynman et al. 1989, 2: 28–6) (22) where α and ƛ depend upon the shape of the partic le and the c harge distribution. Like the last term written explic itly, all later terms go to z ero as a goes to z ero. However, the first term blows up in that limit. So, we c annot by this means define the self-forc e on a point-partic le. Essentially the same problem with the limit would fac e us if we had, instead, assumed that the ac c elerated c harge gives rise to an advanc ed field, but in that c ase, we would get (23) for the self-force. Sinc e Dirac , it has been c ommon to derive an equation of motion for a point-elec tron by assuming that the interac tion of the elec tron with itself is a c ombination of the advanc ed and retarded potentials, spec ific ally, one-half of the differenc e of the advanc ed potential and the retarded potential. If we assume this, the first term whic h is problematic in the limit goes away. Sinc e the later terms all go to z ero in the limit, we are left with (24) as the self-forc e on a point-c harge (where tr   =   2e2 is the time it takes for light to c ross the c lassic al elec tron 3c3m radius). The equation of motion for a c harged point-partic le, known in the nonrelativistic c ase as the Abraham- Lorentz equation, is (25) (See Jackson 1975 for further details.) We will return to this equation later, but for now it is worth noting that the assumptions in the derivation above are not always rejec ted as violating c ausality. For example, Feynman rehearses Dirac 's derivation, and insofar as he is dismissive of it, it is bec ause it c ontains an “arbitrary” assumption not bec ause of any violation of “c ausality” (Feynman et al. 1989, 2–28–5). At any rate, when one derives this equation of motion, there are no imposed initial c onditions, sinc e one is not solving an initial value problem. As suc h, there is no fear of c ontradic ting one's initial c onditions. And, when this is so, it is muc h less c ommon to balk at the inc lusion of the advanc ed potentials even though, presumably, the c ausal interpretation of them as involving bac kward c ausation would remain. This provides evidenc e that when physic ists rejec t the advanc ed potentials as acausal they are not abhorring the c ause c oming after the effec t per se. Rather, they are thinking in terms of an initial value problem where the initial c onditions are already built into the solution to the homogeneous equation and the retarded Green's func tion must be used on pain of c ontradic tion. Further fuel for the c ausal skeptic c omes from realiz ing that various elements within the motivation for representing motion in terms of the retarded Green's func tion are optional. In partic ular, one does not need to build the initial values into the solution to the homogeneous equation. Rather, among other things, one c an also solve a “final value problem” where the state of the system is assigned at a time after the inhomogeneous term has stopped acting. In this case, the very same solution will be represented as (26) where xf (t) is a solution to the homogeneous equation, whic h is determined by a “final c ondition” after the inhomogeneous term bec omes z ero.2 0 In this c ase, so as not to c ontradic t the “final c ondition” one needs to use the advanc ed Green's func tion. But here, exc ept in spec ial c ases, xf (t) will not satisfy the initial c onditions used in f (t)  +   ∫ ∞ adv (t − ′) f( ′)d ′ Page 8 of 24

Causation in Classical Mechanics the previous representation. Only the sum xf (t)  +   ∫ ∞ gadv (t − t′) f(t′)dt′ will satisfy those initial conditions. −∞ But, one c an represent the exac t same solution via the retarded Green's func tion and we are, thus, left without any sense of what it gets wrong. Of c ourse, some might c laim that the advanc ed Green's func tion represents the c ausal direc tionality in the system inc orrec tly, sinc e it depic ts the system as if the c ause c omes after the effec t. This is just to baldly assert that the retarded Green's func tion c orrec tly represents c ausation, sinc e it is the one that adheres to the maxim, “the c ause prec edes the effec t.” However, we have not thereby gained any insight into why we take the maxim to be true and substantive. Why not, on the c ontrary, think that the maxim is (or might be) false or of indeterminate truth value? A c ausal skeptic will c ontinue to wonder here what the grounds are for thinking that the advanc ed Green's func tion inc orrec tly represents the effec t of an impulsive forc e. Moreover, one will wonder why one needs to enter this c ausal morass in the first plac e: one c ould c apture the entire motion of the system via the advanc ed Green's func tion and a suitable solution to the homogeneous equation. So, it is not c lear what one will have missed if one did not get into the business of trying to privilege the retarded Green's func tion in the first plac e. 1.2 The Wave Equation and Spatial Propagation The drama above involving the Green's func tion of the harmonic osc illator c arries over rather straightforwardly to the wave equation. But, the wave equation c ontains additional c omplexities bec ause it is a partial differential equation and, as suc h, involves spatial propagation in addition to mere temporal evolution. For now, c onsider the homogeneous wave equation: (27) Solving this in all of infinite, unbounded spac e involves a pure initial value problem or pure Cauc hy problem. Sinc e it is sec ond order in time, the wave equation needs two initial c onditions: (28) and (29) where x represents all three spatial variables. For the wave equation in all of spac e, these c onditions are suffic ient to yield a unique solution (assuming suffic ient differentiability of the initial data2 1). As suc h, with the wave equation it is not nec essary to impose a “radiation c ondition” at infinity to rule out solutions that involve spheric al waves c ollapsing down onto a point. Rather, a spec ific ation of the de fac to initial c onditions will be suffic ient to rule out waves c ollapsing at a point if, in fac t, there are none. Nature does not need extra c onstraints so as to keep “waves from coming in from infinity.” 1.3 The Wave Equation in a Bounded Spatial Domain There are, of course, other sorts of problems than a pure initial value problem for the wave equation. If one thinks in terms of a bounded spatial domain, one needs to supply boundary c onditions, as well as initial c onditions. There are a variety of different types of boundary c onditions applic able to the wave equation, whic h I will not disc uss in detail here. For eac h type of them, one spec ifies some aspec t of the behavior of the field on the spatial boundary. In a bounded spatial domain, the Green's func tion will be required to solve homogeneous boundary c onditions, meaning that some aspec t of its behavior on the boundary—either its value or the value of its derivative—is set to z ero. One c an use suc h a Green's func tion to give a rather interesting and useful dec omposition of a solution to an initial and boundary-value problem. A solution to the wave equation in bounded regions c an be dec omposed as follows (Barton 1989, 245): (30) (31) Page 9 of 24

Causation in Classical Mechanics (32) (33) (34) (35) where f(x, t) is an inhomogeneous term, g(x, t, x′, t′) is a retarded Green's func tion, t0 is the initial time, and ψ S represents ψ on the spatial boundary surfac e S. Gabriel Barton c alls this rather quaintly the “magic rule” for solving the wave equation, sinc e it allows one to arrive at the solution via quadrature when the Green's func tion is known. One might think of these func tions as follows: ψ f (x, t) gives the dependenc e on the inhomogeneous term (i.e., the forc ing term); it is a solution of the inhomoge-neous equation with homogeneous boundary c onditions and homogeneous initial c onditions; ψ b(x, t) gives the dependenc e on the boundary c onditions; it is a solution of the homogeneous equation with inhomogeneous boundary c onditions and homogeneous initial c onditions; ψ i(x, t) gives the dependenc e on the initial c onditions; it is a solution of the homogeneous equation with homogeneous boundary conditions. The Green's func tion in the magic rule above is the retarded Green's func tion. However, a retrodic tive “magic rule” involving the advanc ed Green's func tion also exists. So, why don't we typic ally represent the solution using the advanc ed Green's func tion? Barton c laims the following: The reason why less attention is paid to retrodic tion than to predic tion is that in general the requisite input information about the future is not available, and that one seldom needs to c onstruc t the past (even though the requisite data are known, referring as they do to the present). (Barton 1989, 257) However, advanc ed Green's func tions are not “unphysic al” in the sense of being useless for the ac c urate desc ription of physic al proc esses. In fac t, in final value problems, one has to use them in the retrodic tive magic rule. A c ausal skeptic c an also note the following: there is room to quibble over whether these func tions represent the way the solution breaks up into c ontributions from the various physic al aspec ts of the system. For example, I desc ribed ψ i (x, t) as giving the dependenc e of the solution on the initial c onditions. But, it is not obvious that this is the right way to think about it. To simplify, suppose that the problem being solved involves homogeneous boundary c onditions of the “Diric hlet variety” (i.e., the spec ific ation that ψ (x,t) vanishes on the boundary) and that there is no forc ing. In this c ase, ψ i (x, t) will be the only c ontribution to the system. Bec ause it involves suc h homogeneous boundary c onditions, waves will reflec t off of the boundary, and suc h reflec tion will be present in ψ i(x, t). As suc h, the statement of dependenc e given above suggests that a wave that is present in the initial c onditions but reflec ts off of the boundary is c aused by the initial c onditions and not by the boundary. But, one c ould plausibly reason in either of the following two ways: (1) With homogeneous boundary c onditions, the boundary is not really doing anything; it is not pumping energy into the system. There is no sourc e present in that sense. The energy that is there in the reflec ted wave c an simply be trac ed bac k to the initial c onditions. So, it is c aused by them and not by the boundary. Thus, the reflec ting homogeneous boundary c onditions really do represent what the initial c onditions c ause c orrec tly: the reflec ted wave is due to the initial c onditions. Or: (2) Were the boundary not there, there would not have been any reflec tion of waves present in the initial data. So, the reflec ted waves that have been attributed to the initial c onditions are attributed inc orrec tly. Rather, they are due to the boundary rather than (or, perhaps, in addition to) the initial c onditions, sinc e it is the boundary that reflec ts the waves. Insofar as it is diffic ult to assess whic h of these two patterns of reasoning is “right,” there is some sense in whic h Page 10 of 24

Causation in Classical Mechanics our “c onc ept of c ause” does not push us to a natural “effec t” of the initial c onditions. But, all of this provides a new reason to wonder whether there is a uniquely c orrec t “c ausal” dec omposition of the field. This reason remains even if we have dec ided that the retarded Green's func tion c orrec tly represents the temporal direc tion of c ausation. Clearly, there is no unique mathematic al dec omposition of the field, but it is not even obvious that there is a unique dec omposition that most nearly reflec ts our thinking about c ausality. But, for all that, we do not seem to be any worse off in terms of understanding elec tromagnetism or other fields of physic s where the wave equation plays a role. 2. Where and Why Do es One Need a Radiatio n Co nditio n? For the wave equation in all of spac e, the initial c onditions suffic e to determine a unique solution. In a bounded domain, the “magic rule” gives the solution direc tly from the applied data involving the forc ing term, initial c onditions, and boundary c onditions. In neither c ase does one need to invoke an additional radiation c ondition so as to selec t a unique solution. There are, however, c ontexts in whic h a radiation c ondition is invoked so as to ensure uniqueness. The most famous suc h c ondition is applied to the Helmholtz equation (whic h will be derived below in several ways) and is c alled the Sommerfeld radiation c ondition: (36) uniformly in all direc tions. What this c ondition does is rule out solutions that involve “inc oming waves,” waves that originate from infinity and propagate inward toward a sourc e. Figure 3.3 Waveguide. Here, I explore where and why suc h a c ondition is needed. Let us start with a simple problem of radiation: 2 2 we imagine a channel starting at z = 0 but having an infinite length along the positive z axis. (See figure 3.3.) We will imagine radiation being pumped into the c hannel via a time-dependent boundary c ondition. Our main equation is, of c ourse, the wave equation (27) in whic h we shall assume that c = 1. In addition, the time-dependent boundary condition along the z = 0 boundary is (37) We also need boundary c onditions along the other sides of the c avity. For simplic ity, we c an assume that it is a perfec tly reflec ting c avity. This is ensured by requiring that (38) along all of those other sides. Moreover, we assume the following initial c onditions at t0 = 0: (39) and (40) Lastly, we assume a “finiteness c ondition” at z = ∞.2 3 This finiteness c ondition, in itself, does not have the c ontent of the Sommerfeld radiation c ondition, sinc e “inc oming waves” c an be finite at infinity. In theory, the c onditions imposed here are enough to yield a unique solution to the problem without any sort of radiation c ondition, but that does not mean that we c an easily find the solution that fits the given data. One typic ally employs (as we did above to find Green's func tions) a “transform” tec hnique of some kind: one solves the problem in the transformed format and then transforms bac k—via an inverse transformation—to the original setting. Page 11 of 24

Causation in Classical Mechanics 2.1 The Laplace Transform Technique One path to the solution is via the Laplac e transform. The Laplac e transform of ψ (x, t) is as follows: (41) where in this subsection ψˆ indicates the Laplace transform of ψ rather than its Fourier transform. To solve the wave equation, we want to Laplac e transform eac h term of it. First, the sec ond time derivative transforms as follows: (42) Next, one transforms the sec ond derivatives of the spatial variables. I show what this amounts to for one spatial variable only, but it obviously works equally for the others: (43) Onc e everything is Laplac e transformed, the wave equation ends up just being (44) But, bec ause of the quiesc ent initial c onditions that we assumed (we are trying to solve the original problem), those last two terms are just z ero. So, one ends up with (45) whic h is known as the “Helmholtz equation” or the “reduc ed wave equation.”2 4 To solve the original problem for the wave equation, one wants to solve the Helmholtz equation with the Laplac e- transformed boundary c onditions and then take the inverse transform so as to get bac k to the desired ψ (x, t), whic h has some time-dependenc e.2 5 When one does this (whic h is not trivial), one sees that ψ(x, t) remains z ero until the sourc e (i.e., the z-boundary behavior) starts. After the sourc e is turned on, there is a wave that travels in the outward z-direc tion but not one that travels in the inward z-direc tion. But, as this simply falls out of the imposed initial and boundary c onditions no “radiation” or “c ausal c ondition” needs to be applied in this c ontext so as to rule out an inc oming wave. In the next sec tion, we shall see when it does have to be applied to it. 2.2 The Fourier Transform Technique Let us now look at the Fourier transform treatment of the c ase. The Fourier transform is (46) where ψˆ is (once again) the Fourier transform of ψ with respect to t. When we take the Fourier transform of the wave equation, we again arrive at the Helmholtz equation, but this time for the Fourier transform: (47) We again solve the Helmholtz equation. After disc arding solutions that do not fit the boundary c onditions and inc luding a time-dependenc e, we get solutions of the form (Snider 2006, 558) (48) where x1 represents the rightmost boundary in the x-direc tion, y1 represents the uppermost boundary in the y- direction, and m and n are the numbers of “wave guide modes.” The first summand of this solution represents an inc oming wave that needs to be eliminated by the Sommerfeld radiation c ondition. So, here is where suc h a radiation condition is needed. Why do we have an inc oming wave in this c ase but not when we used the Laplac e transform? Snider provides a clear answer: Page 12 of 24

Causation in Classical Mechanics [W]hat is the story behind the inc oming wave? And why didn't it appear in the Laplac e Transform? This is best understood by rec alling that the Fourier desc ription is tailored to represent a system for all time, from minus infinity to plus infinity. Its “initial c onditions” c or respond to the system's status at t =−∞ …Now sinc e the waveguide extends from z = 0 to z = −∞, if there were some disturbanc e in the tube at “t = −∞” then by any finite time t its outgoing c omponents would have propagated past every finite point z, but its inc oming c omponents would keep arriv ing (there being no damping mec hanism). This possibility has to be ac c ommodated by the Fourier desc ription … In the Laplac e desc ription we presc ribed quiescent initial c onditions throughout the waveguide at t = 0. This had the effec t of z eroing out suc h “built-in” waves, and no indeterminac y oc c urred in the computations. (Snider 2006, 558–559) In the c ase of the Laplac e transform derivation of the Helmholtz equation, one has the oc c asion to apply the initial c onditions for the problem on the route to the Helmholtz equation. In the Fourier transform derivation, one does not. Thus, one needs some other way to impose the c orrec t initial c onditions for the problem. This is, in effec t, what the Sommerfeld radiation c ondition does. However, in essenc e, the Sommerfeld radiation c ondition has not been invoked so as to adhere to some general princ iple of c ausality. Rather, it has been invoked so as to get the c orrec t solution to the initial value problem that one is solving, a solution that does not have inc oming waves. 2.3 Time Harmonic Waves In addition to the two paths to it given above, the Helmholtz equation results from the wave equation via separation of variables. If we assume that a solution to the wave equation ψ (x, t) is suc h that ψ (x, t) = ψ x(x)ψ t(t), we end up with two func tions: ψx(x), whic h c an be shown to be a solution to the Helmholtz equation and ψ t(t), whic h gives a harmonic time-dependenc e (Zac hmanoglou and Thoe 1986, 267). There are time-harmonic solutions to the wave equation that c ontain only “outgoing waves” but others that c ontain “inc oming waves.” The latter solutions are eliminated by the imposition of the Sommerfeld radiation c ondition.2 6 One thing that is odd about suc h time-harmonic solutions to the wave equation is that throughout all of spac e, waves are present. One c an show that a solution to the Helmholtz equation that is twic e differentiable (i.e., a c lassic al solution to the Helmholtz equation) is analytic (Colton and Kress 1998, 18). So, if a solution to it vanishes in an open subset, it vanishes everywhere. Thus, there are no spatial regions that waves have not reac hed in these time-harmonic solutions to the wave equation. This might make one wonder how suc h time-harmonic solutions to the wave equation relate to solutions arising from certain initial value problems for the wave equation. For, in many initial value problems, one starts with a field that is nonz ero only in a bounded region of spac e. So what is the relation between these two types of solutions? Here is a reasonable suggestion from J. J. Stoker's c lassic book Water Waves: 2 7 A point of view whic h seems to the author reasonable is that the difficulty [in selecting sensible radiation conditions in certain cases where it is unclear what conditions should apply] arises because the problem of determining simple harmonic motions is an unnatural problem in mechanics. One should in princ iple rather formulate and solve an initial value problem by assuming the medium to be originally at rest everywhere outside a suffic iently large sphere, say, and also assume that the periodic disturbanc es are applied at the initial instant and then maintained with a fixed frequenc y. As the time goes to infinity the solution of the initial value problem will tend to the desired steady state solution without the nec essity to impose any but boundedness c onditions at infinity. The steady state problem is unnatural—in the author's view, at least—bec ause a hypothesis is made about the motion that holds for all time, while Newtonian mec hanic s is basic ally c onc erned with the predic tion—in a unique way, furthermore—of the motion of a mec hanic al system from given initial c onditions. Of c ourse, in mec hanic s of c ontinua that are unbounded it is nec essary to impose c onditions at ∞ not derivable direc tly from Newton's laws, but for the initial value problem it should suffic e to impose only boundedness conditions at infinity. (Stoker 1957, 175) Essentially, one thinks of the radiation c ondition as selec ting that solution to the Helmholtz equation that Page 13 of 24

Causation in Classical Mechanics corresponds to the infinite time limit of an initial value problem involving the wave equation where there is fixed- frequenc y periodic forc ing from the initial moment and the field either is initially absent or it starts out c onfined to a bounded region. The problem of finding the solution to the Helmholtz equation c orresponding with a c ertain initial value problem of the forc ed wave equation in the infinite time limit is sometimes c alled the “Princ iple of Limiting Amplitude.” (Tikhonov and Samarskii 1990, 573– 575). From thinking in these terms, one c an see why one wants to eliminate c ertain solutions from the Helmholtz equation by using the radiation c ondition: when one applies a spec ific harmonic time-dependenc e to them, they do not c orrespond with the infinite time limit of the initial value problem to the wave equation that one is solving. So, in essenc e, what is motivating their dismissal is that they violate the long-time behavior assoc iated with the imposed initial c onditions and forc ing. This is easy to lose sight of sinc e, being elliptic and not involving the time variable, the Helmholtz equation does not ac c ept initial c onditions. But, in the end, what one is doing is getting one's solution to the Helmholtz equation (with supplemental harmonic time-dependenc e) to c orrespond with the behavior resulting from the given initial c onditions attac hed to the wave equation in the infinite time limit. In these c ases of time-harmonic wave motions, the radiation c ondition is not justified by appeals to maxims like “the c ause prec edes the effec t.” Rather, the projec t initiated by Stoker is to get away from suc h vague appeals. In pursuit of a projec t that is similar to Stoker's, Wilc ox (1959, 133) starts with the following c laim: Nearly fifty years have passed sinc e Sommerfeld introduc ed his radiation c ondition. During this period it has bec ome c ustomary to use the c ondition in formulating and solving the boundary value problems assoc iated with the diffrac tion of time-harmonic waves. The radiation c ondition is satisfac tory from the mathematical viewpoint in that it leads to boundary value problems having unique solutions. However, the physic al reasons usually advanc ed for adopting it, rather than some other c ondition, are far from c onvinc ing. Our purpose here is to provide a more satisfying foundation for use of Sommerfeld's c ondition by deriving it from other fac ts c onc erning wave propagation that are both mathematic ally demonstrable and evident to physic al intuition. Wilc ox shows that, among other radiation c onditions, Sommerfeld's radiation c ondition is a c onsequenc e of a c ertain property of the solution to an initial and boundary-value problem2 8 for the wave equation in the infinite time limit. Moreover, as Stoker notes, in many c ases when one is dealing with the Helmholtz equation, the path to the right radiation c onditions to impose c omes from thinking in these terms. But, this suggests that the prec ise mathematic al c ontent of suc h radiation c onditions is being driven by solutions to c ertain initial and boundary-value problems. A slogan like “the c ause prec edes the effec t” will not get one to suc h mathematic al c ontent and is, thus, comparatively worthless. Moreover, in well-posed initial and boundary-value problems involving the wave equation, the solution is determined without a c ausality c ondition. Thus, suc h a c ausality c ondition does not seem to be of fundamental importanc e. 3. Backward Causatio n in Po int- Particle Electro dynamics Even if the radiation c ondition is not justified by a brute imposition of the maxim “the c ause prec edes the effec t,” there are other interesting c ases in whic h a c ausality princ iple is frequently c laimed to be invoked. Above I noted Dirac 's derivation of an equation of motion (25) for a c harged point-partic le with self-interac tion. That equation does raise worries among some physic ists though some seem to take it to be an ac c eptable c lassic al equation of motion.2 9 One feature of the equation that one is not totally ac c ustomed to is that it is of the third order, whereas standard c lassic al equations of motion for point-partic les are sec ond order. So, one is c onfronted by a different sort of beast than is usual. Many of the worries about (25) surround the fac t that it allows both runaway solutions (that is, solutions suc h that the ac c eleration grows c ontinually even in the absenc e of external forc es) and pre-ac c elerations (that is, ac c elerations that happen in advanc e of a forc e being applied). Runaway solutions are not in evidenc e in nature. So, one would like to rule them out. The general solution to (25) is (Levine et al. 1977, 75) (49) If one selec ts the initial ac c eleration to be (50) Page 14 of 24

Causation in Classical Mechanics it results formally in the ac c eleration at temporal infinity being z ero, and for runaway solutions the ac c eleration at temporal infinity is not z ero. So, one might motivate imposing this initial ac c eleration by the desire to rule out runaway solutions. By imposing this c ondition, one arrives at a new equation, sometimes c alled the “nonloc al” (in time) equation (Levine et al. 1977, 75; Jac kson 1975, 797), (51) This equation, however, obviously involves pre-ac c eleration insofar as the ac c eleration at t depends upon an integral involving values of the forc e at future times. Thus, a partic le c an start to ac c elerate due to forc es on it in the future. Suc h pre-ac c eleration is sometimes dec lared “unphysic al.” But, this does not give us muc h of a sense of what is wrong with it. Often “unphysic al” just means defying our antecedent expectations as to what ought to happen.3 0 There are many reasons that we would not have expec ted suc h pre-ac c eleration. Nothing in c lassic al mec hanic s would have led us to expec t this, sinc e it is not found in the standard Newtonian equation of motion. But, sometimes suc h pre-ac c eleration is dec lared unphysic al bec ause it represents a violation of c ausality. Some of Mathias Frisc h's c laims that c ausality requirements enter physic al theory refer to this equation and its dismissal by physic ists. For example, he c laims, [A] c ausal interpretation of Dirac 's theory [of c harged point-partic les] also seems to be at the root of the feeling of unease that many physic ists have toward the theory. For the c ausal struc ture of the theory violates several requirements we would like to plac e on c ausal theories. If nothing more were at issue than questions of determination, the nonloc al c harac ter of the equation of motion ought not to be troubling. That is, physic ists themselves appear to be guided by c ausal c onsiderations in their assessment of the theory. (Frisch 2005, 99) Presumably, the idea here is that a “c ausal interpretation” involves the following ideas: (1) Forc es c ause ac c elerations. (2) Insofar as a partic le c an ac c elerate even though it only has forc es on it in the future, one c an see from the nonloc al equation (51) that later forc es are c ausing the ac c eleration. (3) But, that involves pernic ious backward causation, and that explains some of the unease toward the equation. Even if that does c apture how some physic ists are thinking when they rejec t the equation, not all physic ists rejec t it on these grounds. And, even if they all did, that alone is merely a soc iologic al matter whic h does not give us muc h of a feel for whether they are warranted. Reasons as to why suc h a c ausality violation should be partic ularly troublesome are not typic ally given. In some c ases, it is merely c laimed that the equation “violates our ordinary c onc eption of c ausality.” That might be true as a psyc hologic al matter and it might explain some unease, but it gives one no sense as to why one ought to feel unease here. Perhaps our ordinary c onc eption is simply naive. Obviously, physic s has tended to show us that our ordinary c onc eptions (of spac e, of time, of the behavior of the mic rosc opic , etc .) are not partic ularly respec ted by nature. If the dominant worry about (51) were only that it involves bac kward c ausation but nothing c onc rete c ould be said about why that is a genuine worry (other than, perhaps, psyc hologic al fac ts about us), then no one who shares Russell's c ausal skeptic ism needs to be persuaded that there is a legitimate c onstraint that ought to rule out the equation. Moreover, one c ould note that the derivation of the Abraham-Lorentz equation that was given above that ultimately led to (51) assumed bac kward c ausation when it assumed that the field assoc iated with the c harge is a c ombination of the retarded and advanc ed potentials. Presumably, if bac kward c ausation is a c ause for c omplaint, one never should have been willing to assume a premise involving it in the first plac e. One ought to have been antecedently dismissive of the initial steps of the derivation on these grounds. However, as we have seen, Feynman was not. We have yet to see a reason to be. Of c ourse, maybe the bac kward c ausation that appears in the nonloc al equation (51) is somehow worse than the bac kward c ausation openly assumed in the derivation: perhaps not all bac kward c ausation is equally bad. But, if something c an be said about what makes some bac kward c ausation worse than others, then it might be the c ase that it is not really bac kward c ausation per se that is the problem. Rather, it is the feature of the derived equation that makes the bac kward c ausation in it be partic ularly pernic ious. Page 15 of 24

Causation in Classical Mechanics There are, to be sure, other derivations of (25) that work via energy-momentum c onservation and that assume that the field assoc iated with an ac c elerating c harge is fully retarded. (For the two routes, see Poisson (1999). See also the derivation in Parrott (1987, 136– 141).) Someone who prefers derivations that start along those lines c an rejec t the bac kward c ausation of the resulting equation in good faith sinc e, at least, he did not openly and willingly assume it in the derivation. However, one still lac ks any feel for why bac kward c ausation is a real sourc e for c omplaint. And, perhaps, now it will be even harder to say why one ought to rejec t the resulting equation on grounds of c ausality violation: even if one thinks that violation of our ordinary c onc ept of c ausality c reates some presumption (however weak) of falsity, c ertainly a derivation from suc h relatively more sec ure ideas as c onservation of energy might be thought to defeat that presumption.3 1 So, it is not c lear how strong the grounds are for rejec ting (25) or (51) on the basis of the maxim “the c ause prec edes the effec t” alone. This is not to say that there are not troubling c irc umstanc es surrounding (25) and its relativistic variant, the Lorentz -Dirac equation (Rohrlic h 1965, 145): (52) This equation also has pre-ac c eleration solutions and runaway solutions. Moreover, the Lorentz -Dirac equation suggests highly c ounterintuitive behavior that, as far as we know, does not appear in the world. In partic ular, as Eliez er originally proved (Parrott 1987, 198), there are solutions to the Lorentz -Dirac equation ac c ording to whic h a negatively c harged partic le heading toward a positively c harged one will not c ollide with it, but will ultimately be turned away with the negative c harge ac c elerating away from the positive c harge in runaway fashion. As noted in Parrott (1987), this behavior is c ertainly not what one would expec t from two oppositely c harged partic les, whic h would be expec ted to c ollide sinc e opposite c harges attrac t.3 2 Perhaps, then, there are grounds for thinking that this equation is not an appropriate c lassic al equation of motion, but it is not c lear that worry over bac kward c ausation is or should be a major driving forc e in its rejec tion, even if one c hooses to rejec t it as the right c lassic al equation—whic h not all physic ists do. 4. Fro m Causality to Dispersio n Relatio ns As fuel for his anti-Russellian stanc e, Frisc h has rec ently brought up dispersion theory as an arena in whic h c ausality plays a role.3 3 The c ornerstone of dispersion theory is the derivation of “dispersion relations” whic h express the real part of some func tion of a c omplex variable in terms of its imaginary part and vic e versa. Here is how suc h derivations c an go: suppose we start with a Green's func tion that allows us to write the state of a system as follows: (53) We may think of x(t) as the effec t (or “output”) and f(t) as the “c ause” (or “input”). Next, it is typic al to invoke “c ausality,” that the effec t does not start before the c ause. To ensure this, it is required that (54) for t 〈 t′. Bec ause of (54), the Fourier transform of g(τ) (where τ = t − t′) c an be written as (55) The fac t that this integral extends only over the positive reals ensures that g(ω) has an analytic c ontinuation into the upper half of the c omplex plane (i.e., when the imaginary part of ω is greater than z ero).3 4 Onc e that is established, one starts via c onsideration of the following Cauc hy integral (56) over a c ontour C like the one that appears on the right of figure 3.2. Provided that g(ω′) is analytic inside and on the c ontour,3 5 Cauc hy's integral formula assures us (Saff and Snider 2003, 495– 496) that when ω (in the denominator) lies on the c ontour, the following holds: (57) Page 16 of 24