- #1
ralqs
- 99
- 1
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 [itex]E=\frac{1}{2}\dot{x}^2 +\frac{1}{2}x^2 + \frac{\epsilon}{4}x^4[/itex], 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.
For example, suppose the energy E of a one-dimensional system is given by [itex]E=\frac{1}{2}\dot{x}^2 +\frac{1}{2}x^2 + \frac{\epsilon}{4}x^4[/itex], 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.