Basic Theoretical Physics A Concise Overview

48 8 Lagrange Formalism I: Lagrangian and HamiltonianFig. 8.1. Hamilton’s variational principle. The figure shows the ε-dependent setof virtual orbits qv(t) := t + (t2 − 1) + ε · sin (t2 − 1)3, for ε = −0.6, −0.4, . . . , 0.6and times t between t1 = −1 and t2 = 1. The actual orbit, q(t), corresponds to thecentral line (ε = 0) and yields an extremum of the action functional. The virtualorbits can also fan out more broadly from the initial and/or end points than in thisexampleiff (i.e., “if, and only if”) the so-called variational derivative δS := − d ∂L + ∂L δq dt ∂q˙v ∂qvvanishes. An example is shown in Fig. 8.1. The postulate that S is extremal for the actual orbit is called Hamilton’svariational principle of least2 action, and the equations of motion, d ∂L − ∂L = 0 , dt ∂q˙v ∂qvare called Lagrangian equations of the 2nd kind (called “2nd kind” by someauthors for historical reasons). They are the so-called Euler-Lagrange equa-tions3 corresponding to Hamilton’s variational principle. (The more compli-cated Lagrangian equations of the 1st kind additionally consider constraintsand will be treated in a later section.) For the special case where L = m x˙ 2 − V (x) , 2Newton’s equation results. (In fact, the Lagrangian equations of the 2nd kindcan also be obtained from the Newtonian equations by a general coordinatetransformation.) Thus one of the main virtues of the Lagrangian formalismwith respect to the Newtonian equations is that the formalisms are physicallyequivalent; but mathematically the Lagrangian formalism has the essential2 In general, the term “least” is not true and should be replaced by “extremal”.3 Of course any function F (L), and also any additive modification of L by a totalderivative df (q(t),q˙(t),t) , would lead to the same equations of motion. dt

8.4 Cyclic Coordinates; Conservation of Generalized Momenta 49advantage of invariance against general coordinate transformations, whereasNewton’s equations must be transformed from cartesian coordinates, wherethe formulation is rather simple, to the coordinates used, where the formu-lation at first sight may look complicated and very special. In any case, the index v, corresponding to virtual, may be omitted, sincefinally ε ≡ 0. For f ≥ 2 the Lagrangian equations of the 2nd kind are, with i = 1, . . . , f : d ∂L ∂L (8.5) =. dt ∂q˙i ∂qi8.4 Cyclic Coordinates; Conservationof Generalized Momenta; Noether’s TheoremThe quantity ∂L pi := ∂q˙iis called the generalized momentum corresponding to qi. Often pi has thephysical dimension of angular momentum, in the case when the correspondinggeneralized coordinate is an angle. One also calls the generalized coordinatecyclic4, iff ∂L =0. ∂qiAs a consequence, from (8.5), the following theorem5 is obtained.If the generalized coordinate qi is cyclic, then the related generalized mo-mentum ∂L pi := ∂q˙iis conserved.As an example we again consider a spherical pendulum (see Sect. 8.2). Inthis example, the azimuthal angle ϕ is cyclic even if the length l(t) of thependulum depends explicitly on time. The corresponding generalized momen-tum, pϕ = ml2 · sin ϑ · ϕ˙ ,is the z-component of the angular momentum, pi = Lz. In the present case,this is in fact a conserved quantity, as one can also show by elementary argu-ments, i.e., by the vanishing of the torque Dz.4 In general relativity this concept becomes enlarged by the notion of a Killing vector.5 The name cyclic coordinate belongs to the canonical jargon of many centuries and should not be altered.

50 8 Lagrange Formalism I: Lagrangian and Hamiltonian Compared to the Newtonian equations of motion, the Lagrangian formal-ism thus:a) not only has the decisive advantage of optimum simplicity. For suitable coordinates it is usually quite simple to write down the Lagrangian L of the system; then the equations of motion result almost instantly;b) but also one sees almost immediately, because of the cyclic coordinates mentioned above, which quantities are conserved for the system.For Kepler-type problems, for example, in planar polar coordinates we di-rectly obtain the result thatL = M vs2 + m · r˙2 + r2ϕ˙ 2 − V (r) . 2 2The center-of-mass coordinates and the azimuthal angle ϕ are thereforecyclic; thus one has the total linear momentum and the orbital angular mo-mentum as conserved quantities, and because the Lagrangian does not dependon t, one additionally has energy conservation, as we will show immediately. In fact, these are special cases of the basic Noether Theorem, named afterthe mathematician Emmy Noether, who was a lecturer at the University ofGo¨ttingen, Germany, immediately after World War I. We shall formulate thetheorem without proof (The formulation is consciously quite sloppy): The three conservation theorems for (i) the total momentum, (ii) the to-tal angular momentum and (iii) the total mechanical energy correspond (i)to the homogeneity (= translational invariance) and (ii) the isotropy (rota-tional invariance) of space and (iii) to the homogeneity with respect to time.More generally, to any continuous n-fold global symmetry of the system therecorrespond n globally conserved quantities and the corresponding so-calledcontinuity equations, as in theoretical electrodynamics (see Part II). For the special dynamic conserved quantities, such as the above-mentionedRunge-Lenz vector, cyclic coordinates do not exist. The fact that these quan-tities are conserved for the cases considered follows only algebraically usingso-called Poisson brackets, which we shall treat below.8.5 The HamiltonianTo treat the conservation of energy, we must enlarge our context somewhatby introducing the so-called Hamiltonian H(p1, . . . , pf , q1, . . . , qf , t) .This function is a generalized and transformed version of the Lagrangian,i.e., the Legendre transform of −L, and as mentioned below, it has manyimportant properties. The Hamiltonian is obtained, as follows: