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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  1. AMBROSETTI, A. - COTI ZELATI, V., Multiple Homoclinic Orbits for a Class of Conservative Systems. Rend. Sem. Mat. Univ. Padova, vol. 89, 1993, 177-194. Zbl0806.58018MR1229052
  2. BOS, W., Kritische Sehenen auf Riemannischen Elementarraumstücken. Math. Ann., 151, 1963, 431-451. Zbl0113.15204MR159297
  3. COTI ZELATI, V. - RABINOWITZ, P.H., Homoclinic Orbits for second Order Hamiltonian Systems Possessing Superquadratic Potentials. J. Amer. Math. Soc., 4, 1991, 693-727. Zbl0744.34045MR1119200DOI10.2307/2939286
  4. DO CARMO, M.P., Riemannian Geometry. Birkhäuser, Boston1992. Zbl0752.53001MR1138207
  5. GIAMBÓ, R. - GIANNONI, F. - PICCIONE, P., Orthogonal Geodesic Chords, Brake Orbits and Homoclinic Orbits in Riemannian Manifolds. Preprint 2004. Zbl1118.37031
  6. GIAMBÓ, R. - GIANNONI, F. - PICCIONE, P., Multiple brake orbits and homoclinics in Riemannian manifolds. Preprint 2004. Zbl1228.37044
  7. GIANNONI, F. - MAJER, P., On the effect of the domain on the number of orthogonal geodesic chords. Diff. Geom. Appl., 7, 1997, 341-364. Zbl0903.53028MR1485641DOI10.1016/S0926-2245(96)00055-1
  8. GIANNONI, F. - RABINOWITZ, P.H., On the Multiplicity of Homoclinic Orbits on Riemannian Manifolds for a Class of Second Order Hamiltonian Systems. No.D.E.A., 1, 1994, 1-46. Zbl0823.34050MR1273342DOI10.1007/BF01194038
  9. LONG, Y. - ZHU, C., Closed characteristics on compact convex hypersurfaces in R 2 ⁢ n . Ann. of Math., (2), 155, 2002, no. 2, 317-368. Zbl1028.53003MR1906590DOI10.2307/3062120
  10. LIU, C. - LONG, Y. - ZHU, C., Multiplicity of closed characteristics on symmetric convex hypersurfaces in R 2 ⁢ n . Math. Ann., 323, 2002, 201-215. Zbl1005.37030MR1913039DOI10.1007/s002089100257
  11. LUSTERNIK, L. - SCHNIRELMAN, L., Méthodes Topologiques dans les Problèmes Variationelles. Hermann, Paris1934. Zbl0011.02803JFM56.1134.02
  12. MAHWIN, J. - WILLEM, M., Critical Point Theory and Hamiltonian Systems. Springer-Verlag, New York-Berlin1988. Zbl0676.58017
  13. PETUREL, E., Multiple homoclinic orbits for a class of Hamiltonian systems. Calc. Var. PDE, 12, 2001, 117-143. Zbl1052.37049MR1821234DOI10.1007/PL00009909
  14. RABINOWITZ, P.H., Periodic and Eteroclinic Orbits for a Periodic Hamiltonian System. Ann. Inst. H. Poincaré, Analyse Non Linéaire, 6, 1989, 331-346. Zbl0701.58023MR1030854
  15. SERÉ, E., Existence of Infinitely Many Homoclinic Orbits in Hamiltonian Systems. Math. Z., 209, 1992, 27-42. Zbl0725.58017MR1143210DOI10.1007/BF02570817
  16. SERÉ, E., Looking for the Bernoulli Shift. Ann. Inst. H. Poincaré, Analyse Non Linéaire, 10, 1993, 561-590. Zbl0803.58013MR1249107
  17. STRUWE, M., Variational methods. Applications to nonlinear partial differential equations and Hamiltonian systems. 3rd edition, Springer-Verlag, Berlin2000. Zbl0746.49010MR1736116
  18. TANAKA, K., A Note on the Existence of Multiple Homoclinic Orbits for a Perturbed Radial Potential. No.D.E.A., 1, 1994, 149-162. Zbl0819.34032MR1273347DOI10.1007/BF01193949

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