# Is it true that a nondegenerate minimizing periodic orbit of mechanical Hamiltonian system is hyperbolic

Consider mechanical Hamiltonian system of the form $$H(p,q)=\dfrac{\Vert p\Vert^2}{2}+V(q),\quad (q,p)\in T^*\mathbb T^n.$$

Here we suppose the periodic orbit $\gamma$ minimizes the Lagrangian action $$\int_\gamma L(q,\dot q)\,dt$$ locally. The nondegeneracy means that the second variation restricted as a finite dimensional matrix is not degenerate.

If we do not assume the "mechanical", there are many examples having elliptic minimizing periodic orbits.

One example is the Lagrangian periodic orbit in three body problem. For some choice of parameters (mass, eccentricity), the orbit can be elliptic. It is also action minimizing as shown by Venturelli. However, if we reduce the angular momentum conservation, the system is not mechanical.

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