We need to find the period of motion and the regions inaccessible to the particle.
I would like to at least understand how to solve it ideologically, otherwise I have no thoughts at all...
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I think you're going to need to provide more information. It's not clear what you're asking. Are you saying you have a particle moving in the 2D plane and bouncing off the outside of an ellipse? Is the particle bouncing around inside an ellipse? It's not clear what's happening. Can you provide a picture? – John Barber Dec 14 '23 at 22:10
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By "period of motion" are you referring to how long it takes to repeat its path? There are only countably many initial directions the particle cound be travelling that would result in it ever repeating its path. Almost every direction it could be initially travelling will result in a path that will never repeat itself. I suspect the problem had more information that you left out. But that information is probably also necessary to make the problem sensible. – Paul Sinclair Dec 16 '23 at 04:34