MathOverflow is a question and answer site for professional mathematicians. Join them; it only takes a minute:

Sign up
Here's how it works:
  1. Anybody can ask a question
  2. Anybody can answer
  3. The best answers are voted up and rise to the top

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.

share|cite|improve this question

Your Answer


By posting your answer, you agree to the privacy policy and terms of service.

Browse other questions tagged or ask your own question.