Past exam of the mathematics course of the University of Cambridge / 2018 / ib / Paper 3 / 6B / Solution

The Euler-Lagrange equations areThe coordinate $\varphi$ is a cyclic coordinate, so its conjugate momentumis conserved. Substitution into the first equation givesMultiplying by $2\dot{\theta }$ shows thatThus $h$ is conserved. Since $l^2/{\sin }^2\theta ={\sin }^2\theta {\dot{\varphi }}^2$, $h$ is the squared speed on the unit sphere, equivalently twice the kinetic energy.