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Appendix A: Lagrangian Mechanics Appendix A: Lagrangian Mechanics Lagrangian mechanics is not only a very beautiful and powerful formulation of mechanics, but it is also the basis for a deeper understanding of all fundamental interactions in physics. All fundamental equations of motion in physics are encoded in Lagrangian field theory, which is a generalization of Lagrangian mechanics for fields. Furthermore, the connection between symmetries and conservation laws of physical systems is best explored in the framework of the Lagrangian formulation of dynamics, and we also need Lagrangian field theory as a basis for field quantization. Suppose we consider a particle with coordinates x(t) moving in a potential V (x). Then Newton’s equation of motion mx¨ = −∇V (x) is equivalent to the following statement (Hamilton’s principle, 1834): The action integral t1 t1 m S[x]= dt L(x, x˙ )= dt x˙ 2 − V (x) t0 t0 2 is in first order stationary under arbitrary perturbations x(t) → x(t)+δx(t) of the path of the particle between fixed endpoints x(t0)andx(t1) (i.e. the perturbation is only restricted by the requirement of fixed endpoints: δx(t0)= 0andδx(t1) = 0). This is demonstrated by straightforward calculation of the first order variation of S, t1 δS[x]=S[x + δx] − S[x]= dt [mx˙ · δx˙ − δx · ∇V (x)] t0 t1 = − dt δx · (mx¨ + ∇V (x)) . (A.1) t0 Partial integration and δx(t0)=0,δx(t1) = 0 were used in the last step. Equation (A.1) tells us that δS[x] = 0 holds for arbitrary path variation with fixed endpoints if and only if the path x(t) satisfies Newton’s equations, mx¨ + ∇V (x)=0. R. Dick, Advanced Quantum Mechanics: Materials and Photons, 461 Graduate Texts in Physics, DOI 10.1007/978-1-4419-8077-9, c Springer Science+Business Media, LLC 2012 462 A. Lagrangian Mechanics This generalizes to arbitrary numbers of particles (x(t) → xI (t), 1 ≤ I ≤ N), and to the case that the motion of the particles is restricted through con- straints, like e.g. a particle that can only move on a sphere11.Inthecaseof constraints one substitutes generalized coordinates qi(t) which correspond to the actual degrees of freedom of the particle or system of particles (e.g. polar angles for the particle on the sphere), and one ends up with an action integral of the form t1 S[q]= dt L(q,q˙). t0 The function L(q,q˙)istheLagrange function of the mechanical system with generalized coordinates qi(t), and a shorthand notation is used for a mechanical system with N degrees of freedom, (q,q˙)=(q1(t),q2(t),...,qN (t), q˙1(t), q˙2(t),...,q˙N (t)). First order variation of the action with fixed endpoints (i.e. δq(t0)=0, δq(t1) = 0) yields after partial integration t1 ∂L ∂L δS[q]=S[q + δq] − S[q]= dt δqi + δq˙i t0 ∂qi ∂q˙i i i t1 ∂L d ∂L = dt δq − , (A.2) i ∂q dt ∂q˙ t0 i i i where again fixation of the endpoints was used. δS[q] = 0 for arbitrary path variation qi(t) → qi(t)+δqi(t) with fixed endpoints then immediately tells us the equations of motion in terms of the generalized coordinates, ∂L d ∂L − =0. (A.3) ∂qi dt ∂q˙i These equations of motion are called Lagrange equations of the second kind or Euler-Lagrange equations or simply Lagrange equations. The quantity ∂L pi = ∂q˙i is denoted as the conjugate momentum to the coordinate qi. The conjugate momentum is conserved if the Lagrange function depends only on the generalized velocity componentq ˙i but not on qi, dpi/dt =0. Furthermore, if the Lagrange function does not explicity depend on time, we have dL ∂L = piq¨i + q˙i. dt ∂qi 11And it also applies to relativistic particles, see Appendix B. A. Lagrangian Mechanics 463 The Euler-Lagrange equation then implies that the Hamilton function H = piq˙i − L is conserved, dH/dt =0. For a simple example, consider a particle of mass m in a gravitational field g = −gez. The particle is constrained so that it can only move on a sphere of radius r. An example of generalized coordinates are angles ϑ, ϕ on the sphere, and the Cartesian coordinates {X, Y, Z} of the particle are related to the generalized coordinates through X(t)=r sin ϑ(t) · cos ϕ(t), Y (t)=r sin ϑ(t) · sin ϕ(t), Z(t)=r cos ϑ(t). The kinetic energy of the particle can be expressed in terms of the generalized coordinates, m m m K = r˙2 = X˙ 2 + Y˙ 2 + Z˙ 2 = r2 ϑ˙2 +˙ϕ2 sin2 ϑ , 2 2 2 and the potential energy is V = mgZ = mgr cos ϑ. This yields the Lagrange function in the generalized coordinates, m m L = r˙2 − mgZ = r2 ϑ˙2 +˙ϕ2 sin2 ϑ − mgr cos ϑ, 2 2 and the Euler-Lagrange equations yield the equations of motion of the particle, g ϑ¨ =˙ϕ2 sin ϑ cos ϑ + sin ϑ, (A.4) r d ! " ϕ˙ sin2 ϑ =0. (A.5) dt The conjugate momenta ∂L 2 pϑ = = mr ϑ˙ ∂ϑ˙ and ∂L p = = mr2ϕ˙ sin2 ϑ ϕ ∂ϕ˙ are just the angular momenta for rotation in ϑ or ϕ direction. The Hamilton function is the conserved energy p2 p2 H = p ϑ˙ + p ϕ˙ − L = ϑ + ϕ + mgr cos ϑ = K + U. ϑ ϕ 2mr2 2mr2 sin2 ϑ 464 A. Lagrangian Mechanics The immediately apparent advantage of this formalism is that it directly yields the correct equations of motion (A.4,A.5) for the system without ever having to worry about finding the force that keeps the particle on the sphere. Beyond that the formalism also provides a systematic way to identify conservation laws in mechanical systems, and if one actually wants to know the force that keeps the particle on the sphere (which is actually trivial here, but more complicated e.g. for a system of two particles which have to maintain constant distance), a simple extension of the formalism to the Lagrange equations of the first kind can yield that, too. The Lagrange function is not simply the difference between kinetic and po- tential energy if the forces are velocity dependent. This is the case for the Lorentz force. The Lagrange function for a non-relativistic charged particle in electromagnetic fields is m L = x˙ 2 + qx˙ · A − qΦ. 2 This yields the correct Lorentz force law mx¨ = q(E + v × B) for the parti- cle, cf. Section 15.1. The relativistic versions of the Lagrange finction for the particle can be found in equations (B.24,B.25). Direct derivation of the Euler-Lagrange equations for the generalized coordinates qa from Newton’s equation in Cartesian coordinates We can derive the Euler-Lagrange equations for the generalized coordinates of a constrained N-particle system directly from Newton’s equations. This works in the following way: j Suppose we have N particles with coordinates xi ,1≤ i ≤ N,1≤ j ≤ 3, moving in a potential V (x1...N ). The Newton equations d ∂ m x j V x ( i ˙ i )+ j ( 1...N )=0 dt ∂xi can be written as d ∂ ∂ 1 − 2 − j j mkx˙ k V (x1...N ) =0, (A.6) dt ∂x˙ i ∂xi 2 k or equivalently d ∂ ∂ − L x , x˙ , j j ( 1...N 1...N )=0 (A.7) dt ∂x˙ i ∂xi with the Lagrange function 1 L(x , x˙ )= m x˙ 2 − V (x ). (A.8) 1...N 1...N 2 i i 1...N i A. Lagrangian Mechanics 465 If there are C holonomic constraints on the motion of the N-particle system, we can describe its trajectories through 3N − C generalized coordinates qa, 1 ≤ a ≤ 3N − C: j j xi = xi (q,t). (A.9) j Note that in general xi (q,t) will implicitly depend on time t through the time dependence of the generalized coordinates qa(t), but it may also explicitly depend on t because there may be a time dependence in the C constraints. (A simple example for the latter would e.g. be a particle that is bound to a sphere with time-dependent radius R(t), where R(t) is a given time-dependent function.) The velocity components of the system are dx j ∂x j ∂x j x˙ j = i = q˙ i + i . (A.10) i dt a ∂q ∂t a a This implies in particular the equations ∂x˙ j ∂x j i = i (A.11) ∂q˙a ∂qa and j 2 j 2 j ∂x˙ i ∂ xi ∂ xi = q˙b + . (A.12) ∂qa ∂qa∂qb ∂t∂qa b Substitution of (A.11)into(A.12) yields j 2 j 2 j ∂x˙ i ∂ x˙ i ∂ x˙ i = q˙b + . (A.13) ∂qa ∂qb∂q˙a ∂t∂q˙a b Equation (A.11) also yields ∂2x˙ j i =0, ∂q˙a∂q˙b and this implies with (A.13) j 2 j 2 j j d ∂x˙ i ∂ x˙ i ∂ x˙ i ∂x˙ i = q˙b + = . (A.14) dt ∂q˙a ∂qb∂q˙a ∂t∂q˙a ∂qa b With these preliminaries we can now look at the following linear combinations of the Newton equations (A.7): ∂x j d ∂L ∂L i · − =0. ∂q dt ∂x˙ j ∂x j i,j a i i 466 A. Lagrangian Mechanics Insertion of equations (A.11,A.14) yields ∂x˙ j d ∂L ∂x j ∂L d ∂x˙ j ∂x˙ j ∂L i − i + i − i =0, ∂q˙ dt ∂x˙ j ∂q ∂x j dt ∂q˙ ∂q ∂x˙ j i,j a i a i a a i or after combining terms: d ∂x˙ j ∂L ∂L i − =0.
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