MathOverflow is a question and answer site for professional mathematicians. Join them; it only takes a minute:

Sign up
Here's how it works:
  1. Anybody can ask a question
  2. Anybody can answer
  3. The best answers are voted up and rise to the top

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

share|cite|improve this question

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.

share|cite|improve this answer
Thanks, maybe it's hard to give general sufficient properties of $V$. – flavio May 9 '12 at 7:38

Your Answer


By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.