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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.:

Which to me sounds like thatFor energyE > max(U)and(m, n) ∈ ℤ, there exists a periodic motion with this total energy for which the first link of the double pendulum rotates^{2}mtimes and the secondntimes.

__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?