Double pendulum motion (and Lyusternik-Fet Theorem)

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  • Thread starter Lo Scrondo
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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?
 

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