Suppose we have a Hamiltonian system in which the energy is conserved. (Specifically I am looking at a simple double pendulum.) Then we can consider the energy surface $M_e=\{(q,p)\in T^*M\mid H(q,p)=E\}$. For a system with $n$ degrees of freedom this is a $(2n-1)$-dimensional hypersurface of the cotangent bundle.
Everywhere I read about this surface it is assumed to be compact. However it is not obvious to me at all why it should be. Is it always compact, specifically in the case of the double pendulum in a gravitational field?
Edit: I would also be very interested to know whether this has anything to do with the Poincare recurrence theorem and again whether it specifically applied to the double pendulum.
To check that the energy surface is bounded you only have to prove that $H(p,q) = E$ implies that $p,q$ are bounded. If you have a given $H$ this is usually direct.
In the case of the double pendulum the total energy is given by $ E= \frac{1}{2} m \left ( v_1^2 + v_2^2 \right ) + \frac{1}{2} I \left ( {\dot \theta_1}^2 + {\dot \theta_2}^2 \right ) + m g \left ( y_1 + y_2 \right ) $ (https://en.wikipedia.org/wiki/Double_pendulum). If you just rewrite this in terms of $\theta_1, \theta_2$ and their corresponding momenta the result will be direct.