1 Coordinates, Symmetry, and Conservation Laws in Classical Mechanics
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1 Benjamin Good February 25, 20 Coordinaes, Symmery, and Conservaion Laws in Classical Mechanics This documen explores he relaionship beween coordinae changes, symmery, and quaniies ha are conserved during he ime evoluion of classical sysems. Bu before we can examine his relaionship in deail, we mus firs develop a means for describing he dynamics of a general classical sysem. A Lagrangian formulaion of he dynamics will prove o be a paricularly naural way o elucidae hese connecions.. Hamilon s Principle We begin by considering a sysem consising of finiely many coninuous degrees of freedom, and we assume ha he sae of our sysem can be described by N coordinaes q,..., q n which we can denoe in compac form by he vecor q. The behavior of our sysem as i evolves hrough ime can hen be compleely described by he funcion q( (q (,..., q N (, which represens a pah hrough his configuraion space. We now specify he dynamics of our sysem. We assume ha he knowledge of q( and ( dq/d a any paricular ime compleely characerizes he pah of he sysem for all fuure and pas imes. Furhermore, we assume ha here exiss a funcion L(q,, such ha he acion funcional S L(q(, (, d ( is saionary along he acual pah followed by he sysem. This saionariy condiion (known as Hamilon s principle is an elegan and compac way o specify he dynamics of he sysem. However, in is curren form i canno easily be invered o find q(. The calculus of variaions les us ransform his inegral saemen ino a se of differenial equaions for q(, which are much easier o solve (or a he very leas, simulae. The ransformaion proceeds as follows. We begin by calculaing he acion for an alernae pah q ( q( δq( ha differs from he rue pah by an infiniesimal perurbaion δq( ha vanishes a and 2. (The perurbaion in he velociy is simply given by ( ( + d δq(. Since he acion is saionary for he rue d pah, he firs order variaion of S mus vanish: δs L(q (, (, d L(q(, (, d 0 (2 This noaion is simply for convenience, and does no imply ha q ransforms as a physical vecor.
2 We can hen calculae δs in erms of δq: δs L (q + δq, + dd δq, L(q,, d δq i + d q i q i d δq i d ( d δq i d + d [ d ( q i d δq i d [ q i d d ( δq i d where he firs erm in he second o las line vanishes because he perurbaions vanish a he endpoins. Since he perurbaion is effecively arbirary, δs vanishes if and only if d ( 0, i,..., N (3 q i d Coninuous Sysems Coninuous sysems generalize he finie sysems considered above o he case where an inifnie number of degrees of freedom can be indexed by poins x in a coninuous space. In his case, we would wrie q i ( q(x,, bu according o radiion we ypically use he symbol φ(x, and refer o his collecion of degrees of freedom as a field. In addiion o ime derivaives φ(x, φ, we now have spaial derivaives i φ for i,..., d where d is he dimension of he vecor x. We assume a cerain degree of addiiviy in he lagriangian for his sysem, so ha i can be wrien as an inegral of a lagrangian densiy L ha is a funcion only of he fields, heir firs derivaives (spaial and ime, and spaceime: L L(φ, α φ, x α d d x The acion is hen given by S L d L(φ, α φ, x α d d+ x A derivaion similar o he one given above for finie sysems yields he differenial equaions: φ α ( α φ 0 (4 2
3 Mixed discree-coninuous sysems Obviously, we can have a discree number of coninuous fields φ(x, as well, and he Euler-Lagrange equaions become α φ i ( α φ i 0, i,..., N (5.2 Coordinae Transformaions In he previous secion, we assumed ha a Lagrangian funcion L(q,, exiss for each se of coordinae se (q,,, bu we did no say how we would consruc L in general. This raises he naural quesion: if we know he form of he Lagrangian in one frame of reference, how do we consruc he Lagrangian in oher frames of reference? We assume ha we know he Lagriangian for he generalized coordinaes q, and we seek he Lagrangian for anoher se of generalized coordinaes q relaed by he bijecion F (, q G(q, (6 whose inverses are denoed by Wih hese definiions, we find ha f(, q g(q, q( g(q (F (, F (, i ( g F q i i + g F bu by he inverse funcion heorem we also have F/ (/. Then by a sraighforward applicaion of he change of variables heorem, he acion can be wrien as an inegral over : ( S L g(q,, g ( q + g (, f( d (7 F ([, 2 ] Thus, we see ha he Lagrangian in his new coordinae sysem is given by ( L (q (, (, L g(q,, g ( q + g (, f( (8 Noe: The coordinae ransformaions we consider here are ofen referred o as passive ransformaions in he physics lieraure because he sysem iself is unchanged during he ransformaion only he inernal coordinae sysem of he observer is modified. 3
4 Coninuous Sysems This reamen can easily be exended o coninuous sysems, wih and x F (x φ G(φ, x x f(x φ g(φ, x φ(x g(φ (F (x, F (x, α φ i ( j g i ( β φ j ( α F β + ( β g i ( α F β Again, by applying he change of variables heorem, we see ha S d d xl (φ, α φ, x (9 wih F (V L (φ (x, α φ (x, x L (g(φ (x, x, ( j g i ( β φ j ( α F β + ( β g i ( α F β, f(x dedf (0.3 Symmeries and Conservaion Laws Symmery We say ha a coordinae ransformaion represens a symmery of he sysem if no experimen can be done o differeniae beween he wo coordinae sysems. In oher words, he laws of physics (i.e., he equaions of moion for he sysem are he same. This occurs when he ransformed Lagrangian is equal o he original Lagrangian up o an addiive facor of a oal ime derivaive of a scalar funcion: ( L(q (, q, L g(q,, g ( q + g (, f( (, +dh(q d ( I urns ou ha his symmery relaion encodes valuable informaion abou he properies of he sysem. In he case of coninuous symmeries, his relaion will enable us o discover non-rivial conserved quaniies i.e., combinaions of he generalized coordinaes whose values are conserved during he evoluion of he sysem. Coninuous Symmery A coninuous symmery is a coordinae ransformaion ha can be parameerized by a coninuous parameer λ such ha he derivaives / λ and g/ λ exis and lim λ 0 f(, lim g(q, q λ 0 4
5 Conservaion Laws For any coninuous symmery ransformaion, we can ake he derivaive of boh sides of Eq. ( wih respec o λ and he lef hand side rivially vanishes. This yields [ ( 0 d L g(q,, g ( dλ q + g (, f( ] ( { [ ( L 2 f g i + λ q i λ + ( 2 g i q j λ j g ( 2 ( i 2 f q j j λ ( ( 2 g i g ( 2 ( ] } i 2 f + λ λ λ Our assumpions on he coninuous naure of he symmery yield some helpful simplificaions. In paricular, we see ha by inerchanging differeniaion and limis as λ 0, he following ideniies appear: g, g q 0 These ideniies yield a simplified version of he previous equaion: ( 2 f 0 L + g i λ q i λ + [( ( ] 2 g i 2 f q j λ j j + 2 g i λ λ + λ A his poin, we use he Euler-Lagrange equaions o replace / q i wih oal ime derivaives of /. We hen see ha he enire righ hand side can be wrien as he oal ime derivaive of a scalar funcion: d d [ g i λ λ ( i L 0 (2 which implies ha dynamic variable inside he square brackes is a conserved quaniy! Coninuous Sysems.4 Appendix A: Change of variables heorem Add i 5
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