Define the Poisson bracket $\{f, g\}$ between two functions $f\left(q_{a}, p_{a}\right)$ and $g\left(q_{a}, p_{a}\right)$ on phase space. If $f\left(q_{a}, p_{a}\right)$ has no explicit time dependence, and there is a Hamiltonian $H$, show that Hamilton's equations imply

$$\frac{d f}{d t}=\{f, H\} .$$

A particle with position vector $\mathbf{x}$ and momentum $\mathbf{p}$ has angular momentum $\mathbf{L}=\mathbf{x} \times \mathbf{p}$. Compute $\left\{p_{a}, L_{b}\right\}$ and $\left\{L_{a}, L_{b}\right\}$.