Past exam of the mathematics course of the University of Cambridge / 2019 / ii / Paper 1 / 8E / a / Solution

Letbe the infinitesimal generator of the continuous symmetry, and letbe the canonical momentum. Noether theorem states that along every trajectory satisfying the Euler-Lagrange equations,is conserved.

Indeed, differentiate the invariance of $L$ with respect to $s$ at zero:On an Euler-Lagrange trajectory, $\partial L/\partial q_i=d{p}_i/dt$, soThis proves the Noether conserved quantity for a mechanical point symmetry.