Consider a one-dimensional dynamical system with generalized coordinate and momentum $(q, p)$.

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

(b) Find the Poisson brackets $\{q, q\},\{p, p\}$ and $\{q, p\}$.

(c) Assuming Hamilton's equations of motion prove that

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

(d) State the condition for a transformation $(q, p) \rightarrow(Q, P)$ to be canonical in terms of the Poisson brackets found in (b). Use this to determine whether or not the following transformations are canonical:

(i) $Q=\sin q, P=\frac{p-a}{\cos q}$,

(ii) $Q=\cos q, P=\frac{p-a}{\sin q}$,

where $a$ is constant.