Past exam of the mathematics course of the University of Cambridge / 2025 / ib / Paper 3 / 4C (Variational Principles)

The Euler–Lagrange equations areFor $\omega \ne 0$ their general solution iswhile for $\omega =0$, $x=c_{1}t+c_2$.

The action has continuous translation symmetry in $x$; it is also invariant up to endpoint terms under adding homogeneous Jacobi solutions $at+b$ to $x$ and $a\cos t+b\sin t$ to $y$. For $\omega =0$ it additionally has continuous time-translation symmetry; for nonzero $\omega$ only the corresponding discrete period remains.

A complete set of four independent first integrals isIn particular $({\dot{y}}^2+y^2)/2=(C_3^2+C_4^2)/2$ is conserved; when $\omega =0$, the usual total energy is conserved as well.