On the multiplicity of brake orbits and homoclinics in Riemannian manifolds

Roberto Giambò; Fabio Giannoni; Paolo Piccione

Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti Lincei. Matematica e Applicazioni (2005)

  • Volume: 16, Issue: 2, page 73-85
  • ISSN: 1120-6330

Abstract

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Let M , g be a complete Riemannian manifold, Ω ⊂ M an open subset whose closure is diffeomorphic to an annulus. If ∂ ⁡ Ω is smooth and it satisfies a strong concavity assumption, then it is possible to prove that there are at least two geometrically distinct geodesics in Ω ¯ = Ω ⁢ ⋃ ∂ ⁡ Ω starting orthogonally to one connected component of ∂ ⁡ Ω and arriving orthogonally onto the other one. The results given in [5] allow to obtain a proof of the existence of two distinct homoclinic orbits for an autonomous Lagrangian system emanating from a nondegenerate maximum point of the potential energy, and a proof of the existence of two distinct for a class of Hamiltonian systems. Under a further symmetry assumption, it is possible to show the existence of at least dim ⁢ M pairs of geometrically distinct geodesics as above, brake orbits and homoclinics.

How to cite

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Giambò, Roberto, Giannoni, Fabio, and Piccione, Paolo. "On the multiplicity of brake orbits and homoclinics in Riemannian manifolds." Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti Lincei. Matematica e Applicazioni 16.2 (2005): 73-85. <http://eudml.org/doc/252403>.

@article{Giambò2005,
abstract = {Let $(M,g)$ be a complete Riemannian manifold, $\Omega \subset M$ an open subset whose closure is diffeomorphic to an annulus. If $\partial \Omega$ is smooth and it satisfies a strong concavity assumption, then it is possible to prove that there are at least two geometrically distinct geodesics in $\overline\{\Omega\} = \Omega \bigcup \partial \Omega$ starting orthogonally to one connected component of $\partial \Omega$ and arriving orthogonally onto the other one. The results given in [5] allow to obtain a proof of the existence of two distinct homoclinic orbits for an autonomous Lagrangian system emanating from a nondegenerate maximum point of the potential energy, and a proof of the existence of two distinct for a class of Hamiltonian systems. Under a further symmetry assumption, it is possible to show the existence of at least $\text\{dim\}(M)$ pairs of geometrically distinct geodesics as above, brake orbits and homoclinics.},
author = {Giambò, Roberto, Giannoni, Fabio, Piccione, Paolo},
journal = {Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti Lincei. Matematica e Applicazioni},
keywords = {Brake orbits; Homoclinics; Variational methods; brake orbits; homoclinics; variational methods},
language = {eng},
month = {6},
number = {2},
pages = {73-85},
publisher = {Accademia Nazionale dei Lincei},
title = {On the multiplicity of brake orbits and homoclinics in Riemannian manifolds},
url = {http://eudml.org/doc/252403},
volume = {16},
year = {2005},
}

TY - JOUR
AU - Giambò, Roberto
AU - Giannoni, Fabio
AU - Piccione, Paolo
TI - On the multiplicity of brake orbits and homoclinics in Riemannian manifolds
JO - Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti Lincei. Matematica e Applicazioni
DA - 2005/6//
PB - Accademia Nazionale dei Lincei
VL - 16
IS - 2
SP - 73
EP - 85
AB - Let $(M,g)$ be a complete Riemannian manifold, $\Omega \subset M$ an open subset whose closure is diffeomorphic to an annulus. If $\partial \Omega$ is smooth and it satisfies a strong concavity assumption, then it is possible to prove that there are at least two geometrically distinct geodesics in $\overline{\Omega} = \Omega \bigcup \partial \Omega$ starting orthogonally to one connected component of $\partial \Omega$ and arriving orthogonally onto the other one. The results given in [5] allow to obtain a proof of the existence of two distinct homoclinic orbits for an autonomous Lagrangian system emanating from a nondegenerate maximum point of the potential energy, and a proof of the existence of two distinct for a class of Hamiltonian systems. Under a further symmetry assumption, it is possible to show the existence of at least $\text{dim}(M)$ pairs of geometrically distinct geodesics as above, brake orbits and homoclinics.
LA - eng
KW - Brake orbits; Homoclinics; Variational methods; brake orbits; homoclinics; variational methods
UR - http://eudml.org/doc/252403
ER -

References

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