Advances in Differential Equations

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

Roberto Giambò, Fabio Giannoni, and Paolo Piccione

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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

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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.

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