Connecting orbits of time dependent Lagrangian systems

Patrick Bernard[1]

  • [1] Université Joseph Fourier, Institut Fourier, BP 74, 38402 Saint-Martin d'Hères Cedex (France)

Annales de l’institut Fourier (2002)

  • Volume: 52, Issue: 5, page 1533-1568
  • ISSN: 0373-0956

Abstract

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We generalize to higher dimension results of Birkhoff and Mather on the existence of orbits wandering in regions of instability of twist maps. This generalization is strongly inspired by the one proposed by Mather. However, its advantage is that it contains most of the results of Birkhoff and Mather on twist maps.

How to cite

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Bernard, Patrick. "Connecting orbits of time dependent Lagrangian systems." Annales de l’institut Fourier 52.5 (2002): 1533-1568. <http://eudml.org/doc/116017>.

@article{Bernard2002,
abstract = {We generalize to higher dimension results of Birkhoff and Mather on the existence of orbits wandering in regions of instability of twist maps. This generalization is strongly inspired by the one proposed by Mather. However, its advantage is that it contains most of the results of Birkhoff and Mather on twist maps.},
affiliation = {Université Joseph Fourier, Institut Fourier, BP 74, 38402 Saint-Martin d'Hères Cedex (France)},
author = {Bernard, Patrick},
journal = {Annales de l’institut Fourier},
keywords = {connecting orbits; lagrangian systems; minimizing orbits; Lagrangian systems},
language = {eng},
number = {5},
pages = {1533-1568},
publisher = {Association des Annales de l'Institut Fourier},
title = {Connecting orbits of time dependent Lagrangian systems},
url = {http://eudml.org/doc/116017},
volume = {52},
year = {2002},
}

TY - JOUR
AU - Bernard, Patrick
TI - Connecting orbits of time dependent Lagrangian systems
JO - Annales de l’institut Fourier
PY - 2002
PB - Association des Annales de l'Institut Fourier
VL - 52
IS - 5
SP - 1533
EP - 1568
AB - We generalize to higher dimension results of Birkhoff and Mather on the existence of orbits wandering in regions of instability of twist maps. This generalization is strongly inspired by the one proposed by Mather. However, its advantage is that it contains most of the results of Birkhoff and Mather on twist maps.
LA - eng
KW - connecting orbits; lagrangian systems; minimizing orbits; Lagrangian systems
UR - http://eudml.org/doc/116017
ER -

References

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  3. A. Fathi, Book 
  4. A. Fathi, J. Mather, Failure of convergence of the Lax-Oleinik semi-group in the time periodic case, Bull. Soc. Math. France 128 (2000), 473-483 Zbl0989.37035MR1792479
  5. R. Mañé ; G. Contreras, J. Delgado, R. Iturriaga, Lagrangian flows: The dynamics of globally minimizing orbits ; Lagrangian flows: the dynamics of globally minimizing orbits. II, Bol. Soc. Bras. Mat 28 (1997), 141-153 ; 155-196 Zbl0892.58065MR1479499
  6. D. Massart, Aubry set and Mather's action functional, Preprint (2001) Zbl1032.37045
  7. J. Mather, Destruction of invariant circles, Erg. The. and Dyn. Syst 8 (1988), 199-214 Zbl0688.58024MR967638
  8. J. Mather, Differentiability of the minimial average action as a function of the rotation number, Bol. Soc. Bras. Math 21 (1990), 59-70 Zbl0766.58033MR1139556
  9. J. Mather, Variational construction of orbits of twist diffeomorphisms, J. Amer. Math. Soc 4 (1991), 207-263 Zbl0737.58029MR1080112
  10. J. Mather, Action minimizing invariant measures for positive definite Lagrangian systems, Math. Z 207 (1991), 169-207 Zbl0696.58027MR1109661
  11. J. Mather, Variational construction of connecting orbits, Ann. Inst. Fourier (1993) Zbl0803.58019MR1275203
  12. J. Mather, G. Forni, Action minimizing orbits in Hamiltonian systems, Transition to chaos in classical and quantum mechanics 1589 (1994), Springer Zbl0822.70011
  13. J. Moser, Monotone twist Mappings and the Calculs of Variations, Ergodic Theory and Dyn. Syst 6 (1986), 401-413 Zbl0619.49020MR863203
  14. J.-M. Roquejoffre, Convergence to steady states or periodic solutions in a class of Hamilton-Jacobi equations, J. Math. Pures Appl 80 (2001), 85-104 Zbl0979.35033MR1810510

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