Some perturbation results for nonlinear problems (Q1327570)

A Hamiltonian system is considered whose Hamiltonian is a combination of a quadratic polynomial corresponding to the linear equations of motion, and a small higher-order part which is treated as a small perturbation. For this system, a brake orbit is defined as an orbit on which the velocity vanishes twice (at two different moments of time). Once the brake orbit is found, one can construct from it a periodic orbit. In this work, a theorem is proven which gives a minimum number of the brake orbits on a surface with a fixed value of energy. Generally speaking, the minimum number is equal to the number of degrees of freedom of the system. The theorem is proven by reducing it to a theorem giving a minimum number of closed geodesics on a sphere with a slightly perturbed metric; in turn, the latter theorem is proven by means of variational methods.