LAUSR

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 $$\left\{0 \right\} \times \mathbb{T}{^n} \subset {\mathbb{T}^{2n}}$$ { 0 } × T n ⊂ T 2 n for even Hamiltonian on $${\mathbb{T}^{2n}}$$ T 2 n 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{ i L 0 ( z 0 )} pairs of nontrivial 1-periodic brake orbits if i L 0 ( z 0 ) > 0 or at least max{− i L 0 ( z 0 ) − n } pairs of nontrivial 1-periodic brake orbits if i L 0 ( z 0 ) < − n . In the end, we give an example to find brake orbits for certain Hamiltonian via this criterion.