How to show that a trajectory is closed?

[guest1234]

Main question in the title.

I did a group work in analytic mechanics about pendulum in rotating reference frame and stumbled upon this one: http://peer.ccsd.cnrs.fr/docs/00/50/17/84/PDF/PEER_stage2_10.1016%252Fj.ijnonlinmec.2008.03.009.pdf. Does this always hold?

The trajectory is closed if and only if the motion is periodic.

In my solution, there are two coordinates with the same frequency (different amplitudes) but asymmetrical in general sense (diff equations that described system were asymmetrical). Will trajectory close regardless of asymmetry?

The "trajectory fills densely an annulus" condition (see next page) is quite enigmatic, too. The angular velocity of inertial reference frame must be in the same order of magnitude as eigenfrequency (hope it's right term for this) of rotating reference frame, otherwise period of precession is too small and trajectory fills a certain region on a plane of motion. I don't see any irrational numbers involved.

Also, I hope to see any references to literature that help to gain insight on questions like that.

edit: I guess that "trajectory fills an annulus" means the same as "trajectory is not closed", so it all boils down to that irrational number.

 

[mfb]

Motion is periodic <=> after some time t, the system returns to its initial state <=> trajectory is closed

They are equivalent, assuming the system itself is time-independent.