I'm taking an integrability course, it's a bit too fresh and my physics a bit rusty. I'm stuggling with an exercise about the Euler top. I'd appreciate a hand with some questions, or a least some tips. Here's the exercise and some of guesses/questions:
Consider a rotating solid boby attached to a fixed point. Without external forces, its Hamiltonian takes the form $H=\frac{J_x^2}{2I_x}+\frac{J_y^2}{2I_y}+\frac{J_z^2}{2I_z}$. The Poisson brackets or the angular momentum read $\{J_j,J_k\}=\varepsilon_{jkl}J_l$.
- Argue that the system is integrable (wrt. the Liouville-integrabilty definition). Is it superintegrable? Maximally superintegrable? My question/guess: Since the Hamiltonian is an integral of motion, it should do the trick. But why is it enough to exhib only one integral of motion if I should find three of them which Poisson-commute, according to Liouville? My guess for the two other questions is that it is not superintegrable and hence not maximally either, but I don't know how to prove it.
- Using the spherical coordinates $\vec{J}=J(\sin\theta\cos\varphi, \sin\theta\sin\varphi,\cos\theta)$, where $J^2=J^2_x+J^2_y+J^2_z$, deduce the Poisson brackets (or equivalently the symplectic structure) for the spherical variables $\theta$ and $\varphi$. Here, I have no idea and I don't really understand what I should do.
Thank you!