Past exam of the mathematics course of the year 2 of the natural sciences course of the University of Cambridge / 2012 / Paper 1 / 9 / i

Take endpoint-preserving variations $y+\epsilon \eta$, with $\eta (a)=\eta (b)=0$. The first variation of the functional is $\delta I={\int }_a^b(f_y\eta +f_{y^{\prime }}{\eta }^{\prime })dx$; integration by parts supplies the differential equation and its conserved quantity below.

The boundary term $[f_{y^{\prime }}\eta {]}_a^b$ vanishes, so stationarity and the fundamental lemma of the calculus of variations give the Euler-Lagrange equation

If $f$ is independent of $y$, then $f_y=0$ and the Euler-Lagrange equation reduces to $d{f}_{y^{\prime }}/dx=0$. Therefore the first integral isThis is the conserved generalized momentum for a cyclic coordinate.