Jump to ContentJump to Main Navigation
The Lazy UniverseAn Introduction to the Principle of Least Action$

Jennifer Coopersmith

Print publication date: 2017

Print ISBN-13: 9780198743040

Published to Oxford Scholarship Online: June 2017

DOI: 10.1093/oso/9780198743040.001.0001

Show Summary Details
Page of

PRINTED FROM OXFORD SCHOLARSHIP ONLINE (www.oxfordscholarship.com). (c) Copyright Oxford University Press, 2018. All Rights Reserved. Under the terms of the licence agreement, an individual user may print out a PDF of a single chapter of a monograph in OSO for personal use (for details see www.oxfordscholarship.com/page/privacy-policy). Subscriber: null; date: 21 January 2019

(p.222) Appendix A6.3 Energy conservation and the homogeneity of time

(p.222) Appendix A6.3 Energy conservation and the homogeneity of time

The Lazy Universe

Jennifer Coopersmith

Oxford University Press

We consider a case where the Lagrangian has no explicit dependence on time. Then we perform a translation of the time coordinate, tτ, where τ=tϵ, and where ϵ is some small constant. Invariance of the action principle before and after the translation implies that:


(q˙ means d(q)dt, q means d(q)dτ, and Lˉ refers to the transformed Lagrangian).

Next we consider a more general time translation in which ϵ itself is a function of time, ϵ=ϵ(τ). We stipulate that ϵ(τ) is infinitesimal, continuous, and that at the boundaries t is not transformed, in other words, ta=τa and tb=τb, or ϵ(τa)=ϵ(τa)=0.

From t=τ+ϵ(τ) we deduce that d(t)dτ=1+d(ϵ)dτ, and therefore that:


(remembering that is a shorthand for ddτ). Also, we have q˙=d(q)dt=d(q)dτd(τ)dt=q(1+ϵ)1. Therefore, to first order, we have:


Substituting τ for t, and ddτ for ddt, we find that L(qi,q˙i) becomes Lˉ(qi,qi(1ϵ)). As ϵ is small then ϵ also is small and Lˉ may be expanded in a Taylor series expansion in ϵ which, to first order, gives:


(p.223) Also, in the action integral, dt becomes (1+ϵ)dτ. Collecting all these parts together, and ignoring 2nd- and higher-order terms, we finally arrive at:


Now, from the Action Principle, we know that δtatbL(qi,q˙i)dt and δτaτbLˉ(qi,qi)dτ are already guaranteed to be zero. So, we are left with a requirement just involving the last term:


We use our old trick1 - integration by parts - and arrive at:


The first integral contains nothing apart from a total differential, and so it becomes a boundary term:


However, because of our condition, ϵ(τa)=ϵ(τa)=0, then this boundary term is zero. Finally, we are left with just the second integral in (A6.3.7):


(p.224) The integral is preceded by the variation symbol, δ, but what is being varied? It is the infinitesimal function, ϵ, which acts as a ‘variation’, and because this variation is arbitrary (the function ϵ can have any form - provided it is infinitesimal, continuous, and disappears at the boundaries), then it must be the coefficient of ϵ which vanishes. In other words, in order that the integral is zero for arbitrary infinitesimal variations, it is necessary that:


which means that:


Without loss of generality, we may assert this same result in t coordinates:


This conserved quantity has units of energy. No assumptions have been made about L except that it is independent of the time. However, when we consider the simplest default form for L (T is quadratic in the speed coordinates, V depends only on the position coordinates, and L= TV), then it turns out (see Section 6.7, Chapter 6) that Lq˙iq˙iL=E, where E is the total energy.

Thus, assuming the validity of the Principle of Least Action, and assuming the homogeneity of time (the requirement of invariance following a time transformation, even a time-dependent one!), has resulted in the conservation of energy.


(1) (Chapter 4 and the beginning of Chapter 6)