Define an integrable system in the context of Hamiltonian mechanics with a finite number of degrees of freedom and state the Arnold-Liouville theorem.

Consider a six-dimensional phase space with its canonical coordinates $\left(p_{j}, q_{j}\right)$, $j=1,2,3$, and the Hamiltonian

$$\frac{1}{2} \sum_{j=1}^{3} p_{j}^{2}+F(r)$$

where $r=\sqrt{q_{1}^{2}+q_{2}^{2}+q_{3}^{2}}$ and where $F$ is an arbitrary function. Show that both $M_{1}=q_{2} p_{3}-q_{3} p_{2}$ and $M_{2}=q_{3} p_{1}-q_{1} p_{3}$ are first integrals.

State the Jacobi identity and deduce that the Poisson bracket

$$M_{3}=\left\{M_{1}, M_{2}\right\}$$

is also a first integral. Construct a suitable expression out of $M_{1}, M_{2}, M_{3}$ to demonstrate that the system admits three first integrals in involution and thus satisfies the hypothesis of the Arnold-Liouville theorem.