Define a finite-dimensional integrable system and state the Arnold-Liouville theorem.

Consider a four-dimensional phase space with coordinates $\left(q_{1}, q_{2}, p_{1}, p_{2}\right)$, where $q_{2}>0$ and $q_{1}$ is periodic with period $2 \pi$. Let the Hamiltonian be

$$H=\frac{\left(p_{1}\right)^{2}}{2\left(q_{2}\right)^{2}}+\frac{\left(p_{2}\right)^{2}}{2}-\frac{k}{q_{2}}, \quad \text { where } k>0$$

Show that the corresponding Hamilton equations form an integrable system.

Determine the sign of the constant $E$ so that the motion is periodic on the surface $H=E$. Demonstrate that in this case, the action variables are given by

$$I_{1}=p_{1}, \quad I_{2}=\gamma \int_{\alpha}^{\beta} \frac{\sqrt{\left(q_{2}-\alpha\right)\left(\beta-q_{2}\right)}}{q_{2}} d q_{2},$$

where $\alpha, \beta, \gamma$ are positive constants which you should determine.