Is there any example of an autonomous Hamiltonian system with a periodic trajectory isolated in the whole phase space? The Poincar\'e map of such a trajectory within its energy level should be very degenerate, because all the energy levels with close energy values do not contain periodic trajectories: a really weird picture!



Isolated periodic trajectories of Hamiltonian systems
If we take $H=I + \frac13(p^3 -q^3) + I^2(p-q)$ with $\omega =dI\wedge d\theta + dp\wedge dq$,


then


$\dot q = p^2 + I^2$


$\dot p = q^2 + I^2$


$\dot I = 0$


$\dot\theta = 1 + 2I(p-q)$


and the only periodic orbit is for $q=p=I=0$.


