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 $V = \textrm{const.}$).