Past exam of the mathematics course of the University of Cambridge / 2005 / ii / Paper 1 / 15C / i / Solution

Consider a smooth variation $q^a(t)+ϵ{\eta }^a(t)$ with ${\eta }^a(t_1)={\eta }^a(t_2)=0$. Differentiating the action at zero and integrating the velocity term by parts givesThe boundary term vanishes because the endpoints are fixed. Independent compactly supported variations and the fundamental lemma of the calculus of variations make every bracket zero. Thus the Euler-Lagrange equations areThe required principle is stationarity of the action; an extremal need not be a strict minimum.