Multiple brake orbits of even Hamiltonian systems on torus

Abstract

In this paper, we establish a relationship between the Morse index at rest points in the saddle point reduction and the brake-orbit-type Maslov index at corresponding brake orbits. As an application, we give a criterion to find brake orbits which are contractible and start at $$\left\{0 \right\} \times \mathbb{T}{^n} \subset {\mathbb{T}^{2n}}$$ for even Hamiltonian on $${\mathbb{T}^{2n}}$$ by the methods of the Maslov-index theory and a critical point theorem formulated by Bartsch and Wang (1997). Explicitly, if all trivial solutions of a Hamiltonian are nondegenerate in the brake orbit boundary case, there are at least max{iL0 (z0)} pairs of nontrivial 1-periodic brake orbits if iL0 (z0) > 0 or at least max{−iL0 (z0) − n} pairs of nontrivial 1-periodic brake orbits if iL0 (z0) < −n. In the end, we give an example to find brake orbits for certain Hamiltonian via this criterion.