A system is described by the Hamiltonian $H(q, p)$. Define the Poisson bracket $\{f, g\}$ of two functions $f(q, p, t), g(q, p, t)$, and show from Hamilton's equations that

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

Consider the Hamiltonian

$$H=\frac{1}{2}\left(p^{2}+\omega^{2} q^{2}\right)$$

and define

$$a=(p-i \omega q) /(2 \omega)^{1 / 2}, \quad a^{*}=(p+i \omega q) /(2 \omega)^{1 / 2},$$

where $i=\sqrt{-1}$. Evaluate $\{a, a\}$ and $\left\{a, a^{*}\right\}$, and show that $\{a, H\}=-i \omega a$ and $\left\{a^{*}, H\right\}=i \omega a^{*}$. Show further that, when $f(q, p, t)$ is regarded as a function of the independent complex variables $a, a^{*}$ and of $t$, one has

$$\frac{d f}{d t}=i \omega\left(a^{*} \frac{\partial f}{\partial a^{*}}-a \frac{\partial f}{\partial a}\right)+\frac{\partial f}{\partial t}$$

Deduce that both $\log a^{*}-i \omega t$ and $\log a+i \omega t$ are constant during the motion.