Finding if any orbit is closed or not

Question

I came across a problem where I was supposed to answer whether the path was bounded or unbounded and closed or open. I know how to find if it is bounded or not, which can be shown by taking the effective potential diagram vs position coordinate graph and taking points of intersection with the line of total energy. How to find if the orbit is closed or open? Also, if the orbit is closed, will it always be periodic?

Answer 1

You can immediately see if your problem has closed and/or open orbits from the level curves of your Hamiltonian, provided your system is Hamiltonian. If it isn't, you'll need to evaluate your solutions (probably using numeric methods) to see if $\text{solution}(t+\alpha)=\text{solution}(t)$, which is the definition of a closed orbit. Closed and periodic are two names for the same thing, and all closed orbits are bounded.

As an example, take the Simple Harmonic Oscillator. The Hamiltonian is a set of ellipses, which means all orbits are closed. As another example take the Hamiltonian to be

$$H = \frac{p^2}{2} - \frac{q^2}{2} - \frac{q^3}{3} \, .$$

Its level curves around the origin are

You can clearly see a region around $(-1,0)$ that has periodic orbits, together with an homoclinic orbit and many other unbounded, non-periodic trajectories. If your system is non-Hamiltonian, this post in Mathematica.SA might help.

Edit: If you're talking about the two body problem, then every bounded orbit is periodic. This can be seen by solving the equations of motion explicitly, but not by analysing the effective potential. Remember that in the two body problem all orbits are either parabolas or ellipses (or straight lines, which are parabolas of infinite focus). This should clear your view.