# Closed trajectories in phase space

In general, how do you prove that a given trajectory in phase space is closed?

For example, suppose the energy E of a one-dimensional system is given by $E=\frac{1}{2}\dot{x}^2 +\frac{1}{2}x^2 + \frac{\epsilon}{4}x^4$, where ε is a positive constant. Now, I can easily show that all phase trajectories, regardless of energy, are closed by just plotting the various trajectories. But how do I prove it?

I can show that for any (positive) value of E, there's a value of x such that dx/dt is zero. But does this *prove* that the phase trajectories are closed? If it does, I don't see how.

## Answers and Replies

Related Classical Physics News on Phys.org
Matterwave
Yes. Especially because I haven't yet learned the Hamiltonian formalism. I would think that there's a more or less elementary way to show $\dot{x}^2 + x^2 + \frac{\epsilon}{2}x^4 =\mathrm{constant}$ closes on itself, but try as I might I can't come up with anything...