Orthogonal Geodesic Chords, Brake Orbits and Homoclinic Orbits in Riemannian Manifolds

Details Orthogonal Geodesic Chords, Brake Orbits and Homoclinic Orbits in Riemannian Manifolds Orthogonal Geodesic Chords, Brake Orbits and Homoclinic Orbits in Riemannian Manifolds

2004-10-18

arxiv.org/abs/math/0410391v1

Mathematics

Dynamical Systems

24 pages, 2 eps figures, LaTeX

Scientific paper

The study of solutions with fixed energy of certain classes of Lagrangian (or Hamiltonian) systems is reduced, via the classical Maupertuis--Jacobi variational principle, to the study of geodesics in Riemannian manifolds. We are interested in investigating the problem of existence of brake orbits and homoclinic orbits, in which case the Maupertuis--Jacobi principle produces a Riemannian manifold with boundary and with metric degenerating in a non trivial way on the boundary. In this paper we use the classical Maupertuis--Jacobi principle to show how to remove the degeneration of the metric on the boundary, and we prove in full generality how the brake orbit and the homoclinic orbit multiplicity problem can be reduced to the study of multiplicity of orthogonal geodesic chords in a manifold with {\em regular} and {\em strongly concave} boundary.

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