Let $H=H(x, v), x, v \in \mathbb{R}^{n}$, be a smooth real-valued function which maps $\mathbb{R}^{2 n}$ into $\mathbb{R}$. Consider the initial value problem for the equation

$$\begin{aligned} &f_{t}+\nabla_{v} H \cdot \nabla_{x} f-\nabla_{x} H \cdot \nabla_{v} f=0, \quad x, v \in \mathbb{R}^{n}, t>0 \\ &f(x, v, t=0)=f_{I}(x, v), \quad x, v \in \mathbb{R}^{n} \end{aligned}$$

for the unknown function $f=f(x, v, t)$.

(i) Use the method of characteristics to solve the initial value problem, locally in time.

(ii) Let $f_{I} \geqslant 0$ on $\mathbb{R}^{2 n}$. Use the method of characteristics to prove that $f$ remains non-negative (as long as it exists).

(iii) Let $F: \mathbb{R} \rightarrow \mathbb{R}$ be smooth. Prove that

$$\int_{\mathbb{R}^{2 n}} F(f(x, v, t)) d x d v=\int_{\mathbb{R}^{2 n}} F\left(f_{I}(x, v)\right) d x d v$$

as long as the solution exists.

(iv) Let $H$ be independent of $x$, namely $H(x, v)=a(v)$, where $a$ is smooth and realvalued. Give the explicit solution of the initial value problem.