(i) Show that Hamilton's equations follow from the variational principle

$$\delta \int_{t_{1}}^{t_{2}}[p \dot{q}-H(q, p, t)] d t=0$$

under the restrictions $\delta q\left(t_{1}\right)=\delta q\left(t_{2}\right)=\delta p\left(t_{1}\right)=\delta p\left(t_{2}\right)=0$. Comment on the difference from the variational principle for Lagrange's equations.

(ii) Suppose we transform from $p$ and $q$ to $p^{\prime}=p^{\prime}(q, p, t)$ and $q^{\prime}=q^{\prime}(q, p, t)$, with

$$p^{\prime} \dot{q}^{\prime}-H^{\prime}=p \dot{q}-H+\frac{\mathrm{d}}{\mathrm{d} t} F\left(q, p, q^{\prime}, p^{\prime}, t\right)$$

where $H^{\prime}$ is the new Hamiltonian. Show that $p^{\prime}$ and $q^{\prime}$ obey Hamilton's equations with Hamiltonian $H^{\prime}$.

Show that the time independent generating function $F=F_{1}\left(q, q^{\prime}\right)=q^{\prime} / q$ takes the Hamiltonian

$$H=\frac{1}{2 q^{2}}+\frac{1}{2} p^{2} q^{4}$$

to harmonic oscillator form. Show that $q^{\prime}$ and $p^{\prime}$ obey the Poisson bracket relation

$$\left\{q^{\prime}, p^{\prime}\right\}=1$$