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 fist 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 $(m, n) ∈ \mathbb Z^2$, there exists a periodic motion with this total energy for which the first segment 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?