Morse Index Theory For Lagrangian System

In this paper,as first step we study the Morse index theorem for general convex Lagrangian system.Then we compute the difference of those two Morse indices between any self-adjoint and Dirichlet boundary conditions.Consequently,we get a new expression of the relation between Morse index and Maslov index for any self-adjoint boundary condition.Secondly,we consider the Morse index theorem for a very special kind of nonconvex Lagrangian system.At the end,we study the Bott-type index iteration formula for Lagrangian and Hamiltonian system under the equivalent dihedral group action.As applications,we consider the Bott-type index iteration formula and linear instability of a spacelike or timelike closed geodesic in semi-Riemannian manifold,index theorem and strong linear stability of a closed geodesic in Riemannian manifold,the stability problem of brake orbit.