Indices and Stability of the Lagrangian System on Riemannian Manifold

Abstract

In this paper, let m ≥ 1 be an integer, M be an m-dimensional compact Riemannian manifold. Firstly the linearized Poincare map of the Lagrangian system at critical point x $${d \over {dt}}{L_q}\left( {t,x,\dot x} \right) - {L_p}\left( {t,x,\dot x} \right) = 0$$ is explicitly given, then we prove that Morse index and Maslov-type index of x are well defined whether the manifold M is orientable or not via the parallel transport method which makes no appeal to unitary trivialization and establish the relation of Morse index and Maslov-type index, finally derive a criterion for the instability of critical point and orientation of M and obtain the formula for two Maslov-type indices.