Past exam of the probability, statistics and options I course of the University of Oxford / 2010 / G (Calculus of Variations)

Independent endpoint-fixed variations of the two coordinates give the Euler-Lagrange equationsDifferentiating the sphere parametrization gives the Riemannian metric $d{s}^2=a^2(d{\theta }^2+{\sin }^2\theta d{\varphi }^2)$. Its energy functional is the stated $I$. Its stationary paths are geodesics with affine, constant-speed parametrization; conversely a geodesic can be so parametrized. This explains the use of energy rather than arc length.

For $F=a^2({\dot{\theta }}^2+{\sin }^2\theta {\dot{\varphi }}^2)/2$, the equations areThe second gives the conserved generalized momentum ${\sin }^2\theta \dot{\varphi }=h$. Since $F$ has no explicit time dependence, its Beltrami identity gives constant $\dot{\theta }{F}_{\dot{\theta }}+\dot{\varphi }{F}_{\dot{\varphi }}-F=F$. Dividing by $a^2/2$ gives the second first integral,Thus $E$ is the squared speed divided by $a^2$, while $h$ records the rotational symmetry about the polar axis. The coordinate equations apply away from the polar coordinate singularities.