TIME TRANSLATION SYMMETRY AND CONSERVATION OF ENERGY

Link to: physicspages home page. To leave a comment or report an error, please use the auxiliary blog and include the title or URL of this post in your comment. Post date: 8 August 2021. Coleman’s second example of applying Noether’s theorem is with sym- metry under time translation. This time, we consider a general Lagrangian

L = L(qa,q˙a) (1) The only assumption here is that L does not depend explicitly on time. We consider the transformation

qa → qa (t + λ) (2) That is, we translate the time coordinate t by a fixed amount λ for all gen- eralized coordinates. To apply Noether’s theorem to this symmetry, we need to find

∂L ∂L DL = Dqa + Dq˙a (3) ∂qa ∂q˙a ∂L = Dqa + p Dq˙a (4) ∂qa a where

∂L p ≡ (5) a ∂q˙a is the momentum conjugate to the coordinate qa, and

a a ∂q Dq ≡ (6) ∂λ λ=0 d Dq˙a = Dqa (7) dt In this example, 1 TIME TRANSLATION SYMMETRY AND CONSERVATION OF ENERGY 2

a a ∂q Dq = (8) ∂λ λ=0 ∂qa = = q˙a (9) ∂t The last line follows from 2 because t and λ always appear in the combina- tion t + λ, so a derivative with respect to one of t and λ is the same as the derivative with respect to other. To calculate DL in 4 we need the following:

∂L ∂L Dqa = q˙a (10) ∂qa ∂qa ∂ Dq˙a = Dqa (11) ∂t = q¨a (12) so we have

∂L ∂L DL = Dqa + Dq˙a (13) ∂qa ∂q˙a ∂L ∂L = q˙a + q¨a (14) ∂qa ∂q˙a The total derivative of a general Lagrangian L(qa,q˙a,t) with respect to time is, from the chain rule:

dL ∂L ∂L ∂L = q˙a + q¨a + (15) dt ∂qa ∂q˙a ∂t Therefore, we have

∂L ∂L DL = q˙a + q¨a (16) ∂qa ∂q˙a dL ∂L = − (17) dt ∂t However, the Lagrangian we’re considering here has no explicit time de- pendence, so

∂L = 0 (18) ∂t and therefore TIME TRANSLATION SYMMETRY AND CONSERVATION OF ENERGY 3

dL DL = (19) dt This symmetry results in a conservation law if dF DL = (20) dt for some function F (qa,q˙a,t). From 19, we see that

F = L (21) The conserved quantity is Q, defined by

a Q = paDq − F (22) so from 22 we can now find the conserved quantity, using 9:

a Q = paDq − F (23) a = paq˙ − L (24) This quantity is the Hamiltonian, as defined by a Legendre transforma- tion. The Hamiltonian is usually taken to be the energy E of the system, so we see that Noether’s theorem applied to a symmetry under time translation results in energy being the corresponding conserved quantity. Note that the assumption 18 that the Lagrangian has no explicit time de- pendence is crucial in this derivation. If L did have time dependence, we wouldn’t end up with the conserved quantity Q being equal to the energy.

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