1. Consider a Hamiltonian system with coordinates $(\vec{q}, \vec{p}) = (q_1, \dots, q_f; p_1, \dots, p_f)$ with Hamiltonian $H(\vec{q}, \vec{p})$.
(a) Define the Poisson bracket, $[u, v]$, of two functions $u(\vec{q}, \vec{p})$, $v(\vec{q}, \vec{p})$.
(b) An infinitesimal canonical transformation $e\epsilon G$ ($|\epsilon| \ll 1$) may be written as $\delta w = \epsilon[w, G]$ where $w$ and the generator $G$ are also functions of $(\vec{q}, \vec{p})$. Show that the commutator of two such transformations may be written as
$(\delta_1 \delta_2 - \delta_2 \delta_1)w = -\epsilon_1 \epsilon_2[w, [G_1, G_2]]$.
(The Jacobi identity can be assumed, but should be stated.)
(c) Explain how this result implies that if $u$ and $v$ are constants of the motion then so is $[u, v]$.
(d) The Hamiltonian of a spherical top (moment of inertia $A$), pivoted freely at its centre of mass is
$H(\theta, \phi, \psi, p_\theta, p_\phi, p_\psi) = \frac{1}{2A} \left[ p_\theta^2 + \frac{(p_\phi^2 + p_\psi^2 - 2p_\phi p_\psi \cos\theta)}{\sin^2 \theta} \right]$,
where $\theta, \phi, \psi$ are Euler angles. Show that the function
$u = p_\theta + i \frac{(p_\phi \cos\theta - p_\psi)}{\sin\theta} e^{i\phi}$
and its complex conjugate are constants of motion.
(e) Identify $[u, u^*]$.
Can you generate further constants of motion from the Poisson brackets you have already found?
