I am given the Lagrangian for a particle of mass $m$ orbiting a black hole of radius $r_h$:
$$L=\frac{1}{2} m\left\{-F(r) c^{2} \dot{t}^{2}+F(r)^{-1} \dot{r}^{2}+r^{2} \dot{\phi}^{2}\right\}, \quad F(r)=1-\frac{r_{h}}{r}$$
I have derived the momenta to be:
$$\begin{aligned} p_{t} &=-m c^{2}\left(1-r_{h} / r\right) \dot{t} \\ p_{r} &=m\left(1-r_{h} / r\right)^{-1} \dot{r} \\ p_{\phi} &=m r^{2} \dot{\phi} \end{aligned}$$
and the Hamiltonian to be
$$H=\frac{1}{2 m}\left\{-\frac{p_{t}^{2}}{1-r_{h} / r}+\left(1-r_{h} / r\right) p_{r}^{2}+\frac{p_{\phi}^{2}}{r^{2}}\right\}$$
How can one determine which quantities are conserved?
(Answer: The conserved quantities are $p_t$, $p_\phi$ and $H$ — I am unsure how this can be seen from the above).
Via Noether's theorem (thanks, E. Bellec!) and the Euler-Lagrange equations,
$$\frac{\mathrm{d}}{\mathrm{d}t}\left( \frac{\partial L}{\partial \dot{t}} \right) = \frac{\mathrm{d} p_t}{\mathrm{d}t} = 0$$
Hence $p_t$ is a conserved quantity (and similar for $p_\phi$). H has no explicit time dependence so is also conserved. $\square$