Closed trajectories in phase space

[ralqs]

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.

 

[Matterwave]

You could always just use the Hamilton equations to solve for the equations of motion p(t) and x(t) and then solve them for each other to get something like p(x) and show that this results in closed orbits.

Are you asking for a simpler way to do it?

 

[ralqs]

Are you asking for a simpler way to do it?

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...