Double pendulum motion (and Lyusternik-Fet Theorem)

[Lo Scrondo]

Hi everyone!
I recently came across the Lyusternik-Fet theorem concerning closed geodesics on a compact manifold.

For simplicity of description, take the 2-torus, and imagine it represents the configuration space of a double pendulum.
For every pair of integers m, n (where m represents the number of rotations done by the first link and n by the second), there exist a periodic motion that on such torus traces a closed geodesic.

A way in which the Theorem is presented is, e.g.:

For energy E > max(U) and (m, n) ∈ ℤ², there exists a periodic motion with this total energy for which the first link of the double pendulum rotates m times and the second n times.

Which to me sounds like that for every value of the total energy E (provided it's just bigger than the maximal value of the potential energy U) I could get a periodic motion with arbitrary m and n...which seems absurd.

What I haven't understood?

 

[wrobel]

This is a well-know fact from geodesics theory. Do do Lyusternik-Fet? Strange. Theorem IV.5.1
A Modern Introduction
Second Edition
ISAAC CHAVEL