What does it mean to say that a finite-dimensional Hamiltonian system is integrable? State the Arnold-Liouville theorem.

A six-dimensional dynamical system with coordinates $\left(x_{1}, x_{2}, x_{3}, y_{1}, y_{2}, y_{3}\right)$ is governed by the differential equations

$$\frac{\mathrm{d} x_{i}}{\mathrm{~d} t}=-\frac{1}{2 \pi} \sum_{j \neq i} \frac{\Gamma_{j}\left(y_{i}-y_{j}\right)}{\left(x_{i}-x_{j}\right)^{2}+\left(y_{i}-y_{j}\right)^{2}}, \quad \frac{\mathrm{d} y_{i}}{\mathrm{~d} t}=\frac{1}{2 \pi} \sum_{j \neq i} \frac{\Gamma_{j}\left(x_{i}-x_{j}\right)}{\left(x_{i}-x_{j}\right)^{2}+\left(y_{i}-y_{j}\right)^{2}}$$

for $i=1,2,3$, where $\left\{\Gamma_{i}\right\}_{i=1}^{3}$ are positive constants. Show that these equations can be written in the form

$$\Gamma_{i} \frac{\mathrm{d} x_{i}}{\mathrm{~d} t}=\frac{\partial F}{\partial y_{i}}, \quad \Gamma_{i} \frac{\mathrm{d} y_{i}}{\mathrm{~d} t}=-\frac{\partial F}{\partial x_{i}}, \quad i=1,2,3$$

for an appropriate function $F$. By introducing the coordinates

$$\mathbf{q}=\left(x_{1}, x_{2}, x_{3}\right), \quad \mathbf{p}=\left(\Gamma_{1} y_{1}, \Gamma_{2} y_{2}, \Gamma_{3} y_{3}\right),$$

show that the system can be written in Hamiltonian form

$$\frac{\mathrm{d} \mathbf{q}}{\mathrm{d} t}=\frac{\partial H}{\partial \mathbf{p}}, \quad \frac{\mathrm{d} \mathbf{p}}{\mathrm{d} t}=-\frac{\partial H}{\partial \mathbf{q}}$$

for some Hamiltonian $H=H(\mathbf{q}, \mathbf{p})$ which you should determine.

Show that the three functions

$$A=\sum_{i=1}^{3} \Gamma_{i} x_{i}, \quad B=\sum_{i=1}^{3} \Gamma_{i} y_{i}, \quad C=\sum_{i=1}^{3} \Gamma_{i}\left(x_{i}^{2}+y_{i}^{2}\right)$$

are first integrals of the Hamiltonian system.

Making use of the fundamental Poisson brackets $\left\{q_{i}, q_{j}\right\}=\left\{p_{i}, p_{j}\right\}=0$ and $\left\{q_{i}, p_{j}\right\}=\delta_{i j}$, show that

$$\{A, C\}=2 B, \quad\{B, C\}=-2 A$$

Hence show that the Hamiltonian system is integrable.