Advances in Differential Equations

Orthogonal geodesic chords, brake orbits and homoclinic orbits in Riemannian manifolds

Roberto Giambò, Fabio Giannoni, and Paolo Piccione

Full-text: Access denied (no subscription detected)

We're sorry, but we are unable to provide you with the full text of this article because we are not able to identify you as a subscriber. If you have a personal subscription to this journal, then please login. If you are already logged in, then you may need to update your profile to register your subscription. Read more about accessing full-text


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 homoclinics, in which case the Maupertuis--Jacobi principle produces a Riemannian manifold with boundary and with metric degenerating in a nontrivial 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 multiplicity problem can be reduced to the study of multiplicity of orthogonal geodesic chords in a manifold with regular and strongly concave boundary.

Article information

Adv. Differential Equations, Volume 10, Number 8 (2005), 931-960.

First available in Project Euclid: 18 December 2012

Permanent link to this document

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 37J45: Periodic, homoclinic and heteroclinic orbits; variational methods, degree-theoretic methods
Secondary: 58E10: Applications to the theory of geodesics (problems in one independent variable)


Giambò, Roberto; Giannoni, Fabio; Piccione, Paolo. Orthogonal geodesic chords, brake orbits and homoclinic orbits in Riemannian manifolds. Adv. Differential Equations 10 (2005), no. 8, 931--960.

Export citation