G4003 Advanced Mechanics 1. We already saw that if q is a cyclic variable, the associated conjugate momentum is conserved, L = const.

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1 G4003 Avance Mechanics 1 The Noether theorem We alreay saw that if q is a cyclic variable, the associate conjugate momentum is conserve, q = 0 p q = const. (1) This is the simplest incarnation of Noether s theorem, which states that whenever we have a continuous symmetry of Lagrangian, there is an associate conservation law. By symmetry we mean any transformation of the generalize coorinates q, of the associate velocities q, an possibly of the time variable t, that leaves the value of the Lagrangian unaffecte. By continuous symmetry we mean a symmetry with a continuous constant parameter, typically infinitesimal, say ɛ, that we can ial, an that measures how far from the ientity the transformation is bringing us. In a sense ɛ measures the size of the transformation. In the case of the cyclic coorinate iscusse above, the corresponing symmetry is simply q(t) q(t) + ɛ, q(t) q(t), t t, (2) that is, an infinitesimal shift of the cyclic coorinate. Inee, if we perform these replacements in the Lagrangian, at first orer in ɛ the Lagrangian changes by δl L(q + ɛ, q; t) L(q, q; t) q ɛ, (3) which vanishes if an only if q is cyclic. Theorem: Consier a Lagrangian system with n egrees of freeom q 1,..., q n. If for certain functions γ (t) an for constant infinitesimal ɛ the transformation q (t) q (t) + ɛγ (t), q (t) q (t) + ɛ γ (t), t t, (4) is a symmetry, i.e. if it leaves the Lagrangian unaffecte, then the quantity n =1 γ (5) is a constant of motion, i.e. it is conserve. Notice that in (30) we are not transforming the time variable. We will treat the case of time transformations, in particular of time translations, separately below. Proof: By efinition of symmetry, the change in the Lagrangian upon the replacements (30) must vanish δl L(q + ɛγ, q + ɛ γ ; t) L(q, q ; t) = 0. (6) At first orer in ɛ, this equation becomes [ ɛγ + ] ɛ γ = 0. (7) q

2 2 The Noether Theorem We can rewrite the first term by using the equations of motion: We are left with The l.h.s. we can rewrite as a total time erivative =,. (8) q t [ ɛγ + ] ɛ γ = 0. (9) t ɛ t ( This implies that the quantity (5) is conserve: ) γ = 0 (10) γ = const. (11) Examples Eucliean translations. Consier N point particles, interacting via a potential. The Lagrangian is 1 L = m r 2 a 2 a V ( r 1,..., r N ). (12) If the potential only epens on the relative positions r a r b, an not on the absolute ones (i.e., if there are no external forces), V = V ( r 1 r 2,..., r 1 r N, r 2 r 3,... ), (13) then overall translations of the system are a symmetry. An infinitesimal translation of length ɛ in an arbitrary irection ˆn takes the form The corresponing conserve quantity is thus r a r a + ɛˆn, ra r a, t t a. (14) γ = 3 ˆn i = const (15)

3 G4003 Avance Mechanics 3 (the role of the γ s is playe by the cartesian components of ˆn.) The irection ˆn is the same for all particles an we can thus pull it out of the sum over a: 3 ˆn i N ṙ i a = const (16) From (12) we have so that P i = m a ṙ i a (17) ṙ i a is nothing but the i-th component of the total momentum of the system. Our conservation law thus takes the form ˆn P = const. (19) Since ˆn is an arbitrary irection, the whole vector P shoul be constant P = (18) const. (20) We therefore see that the conservation of the total momentum in the absence of external forces is a irect consequence of the invariance of the Lagrangian uner spacial translations. Eucliean rotations. If we make the further assumption that the potential only epens of the mutual istances r a r b between the particles, an not on the orientation of the relative position vectors r a r b, V = V ( r 1 r 2,..., r 1 r N, r 2 r 3,... ), (21) then the Lagrangian is also invariant uner overall rotations of the system, because the potential is, an the kinetic energy is also since it only involves the scalar quantities r a 2. An infinitesimal rotation of angle ɛ about an arbitrary axis ˆn takes the form r a r a + ɛ ˆn r a, ra r a + ɛ ˆn r a, t t a. (22) The corresponing conserve quantity is γ = 3 (ˆn ra ) i = const (23) (the role of the γ s is playe by the cartesian components of (ˆn r a ).) Using (17) we get m a ra (ˆn r a ) = const. (24)

4 4 The Noether Theorem For any three vectors A, B, C, the following ientity hols: We can thus rewrite our conservation law as A ( B C) = B ( C A) (25) ˆn ( r a m a ra ) = const. (26) The irection ˆn is the same for all particles. We can then pull it out of the sum, an we recognize in the remainer the total angular momentum of the system Our conservation law becomes or, since the irection ˆn is arbitrary, L r a m a ra. (27) ˆn L = const, (28) L = const. (29) The conservation of the total angular momentum is a irect consequence of the invariance of the Lagrangian uner overall rotations of the system. Time translations. This case has to be treate separately because our simplifie formulation of the general theorem oes not cover it. By time-translation we mean the transformation q (t) q (t), q (t) q (t), t t + ɛ. (30) That is, we o nothing to the coorinates an to the velocities, but we shift time by an infinitesimal constant. This is a symmetry if an only if the Lagrangian oes not epen explicitly on time. Inee, the variation of the Lagrnagian uner the above transformation woul be δl L(q, q; t + ɛ) L(q, q; t) t ɛ, (31) which vanishes if an only if the partial time-erivative of the Lagrangian vanishes. Recall that the total time-erivative of the Lagrangian oes not vanish in general, because on any given solution the value of Lagrangian epens on time also through the time epenence of the q s an of the q s. One has t L = t + [ q q + q ] We can rewrite the first term insie the brackets via the eom (8). We get t L = t + [ q + ] q t q (32) (33)

5 G4003 Avance Mechanics 5 We then notice that the two terms insie the brackets combine to give a total time erivative: This equation is more conveniently rewritten as where we efine H, the Hamiltonian of the system, as t L = t + q (34) t q t H = t, (35) H q L. (36) In summary, equation (35) is always vali, but if the Lagrangian is invariant uner timetranslations, that is if it oes not epen explicitly on time, then the Hamiltonian of the system is conserve H = const. (37) In most physically relevant cases the value of the Hamiltonian is the total energy. We thus iscovere that the conservation of energy is a irect consequence of the invariance of the Lagrangian uner time translations. Uner stable conitions, if you perform a lab experiment toay or tomorrow you expect to get the same results. This fact alone implies that energy is conserve.