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) Verify the Leibniz rule

$$\{f g, h\}=f\{g, h\}+g\{f, h\}$$

(c) Explain what is meant by a canonical transformation $(q, p) \rightarrow(Q, P)$.

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

(i) $Q=\frac{q^{2}}{2}, P=\frac{p}{q}$,

(ii) $Q=\tan q, P=p \cos q$,

(iii) $Q=\sqrt{2 q} e^{t} \cos p, P=\sqrt{2 q} e^{-t} \sin p$.