Constructing an example of Hamiltonian flow

Question

I have this Hamiltonian flow generated by $$ h(x, \xi ) = V(x) + \sqrt{\xi ^2 + 1}, \quad x, \xi \in \mathbb{R}^3, $$ so the defining equations are $$\begin{align*}\begin{cases} \frac{dx}{dt} &= \frac{\xi (t)}{ \sqrt{\xi ^2 (t) + 1}} \\ \frac{d \xi }{dt} &= - \nabla _x V(x(t)) \end{cases}, \qquad (x(0), \xi (0)) = (x_0, \xi _0). \end{align*}$$ I want to find an example with $V\in C_0^\infty (\mathbb{R}^3)$ (i.e. compactly supported and smooth) such that the solution $(x(t), \xi (t))$ at some energy $E>1$ (i.e. $h(x_0, \xi _0) = E$ and hence $h(x(t), \xi (t)) = E$) satisfies $\lim _{|t|\to \infty }|x(t)| = \infty $.

I'm also grateful for non compactly supported examples (except for $V = \textrm{const.}$).

Answer 1

This is a typical problem of special relativity. The next best choice is a constant acceleration $a$ providing $V(x)=ax$. This has the well-known solution

$$x = d+ \frac{1}{a}\sqrt{1+ a^2 t^2}$$

being $d$ a constant, that is unbounded as required.