(a) State Hamilton's equations for a system with $n$ degrees of freedom and Hamilto$\operatorname{nian} H(\mathbf{q}, \mathbf{p}, t)$, where $(\mathbf{q}, \mathbf{p})=\left(q_{1}, \ldots, q_{n}, p_{1}, \ldots, p_{n}\right)$ are canonical phase-space variables.

(b) Define the Poisson bracket $\{f, g\}$ of two functions $f(\mathbf{q}, \mathbf{p}, t)$ and $g(\mathbf{q}, \mathbf{p}, t)$.

(c) State the canonical commutation relations of the variables $\mathbf{q}$ and $\mathbf{p}$.

(d) Show that the time-evolution of any function $f(\mathbf{q}, \mathbf{p}, t)$ is given by

$$\frac{d f}{d t}=\{f, H\}+\frac{\partial f}{\partial t}$$

(e) Show further that the Poisson bracket of any two conserved quantities is also a conserved quantity.

[You may assume the Jacobi identity,

$$\{f,\{g, h\}\}+\{g,\{h, f\}\}+\{h,\{f, g\}\}=0 .]$$