We consider a case where the Lagrangian has no explicit dependence on time. Then we perform a translation of the time coordinate, , where , and where is some small constant. Invariance of the action principle before and after the translation implies that:
( means , means , and refers to the transformed Lagrangian).
Next we consider a more general time translation in which itself is a function of time, . We stipulate that is infinitesimal, continuous, and that at the boundaries t is not transformed, in other words, and , or .
From we deduce that , and therefore that:
(remembering that is a shorthand for ). Also, we have . Therefore, to first order, we have:
Substituting for t, and for , we find that becomes . As is small then also is small and may be expanded in a Taylor series expansion in which, to first order, gives:
Now, from the Action Principle, we know that and 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, , 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, , but what is being varied? It is the infinitesimal function, , which acts as a ‘variation’, and because this variation is arbitrary (the function can have any form - provided it is infinitesimal, continuous, and disappears at the boundaries), then it must be the coefficient of 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= ), then it turns out (see Section 6.7, Chapter 6) that , 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.