Remote Access Journal of the American Mathematical Society
Green Open Access

Journal of the American Mathematical Society

ISSN 1088-6834(online) ISSN 0894-0347(print)

 
 

 

The dynamics of pseudographs in convex Hamiltonian systems


Author: Patrick Bernard
Journal: J. Amer. Math. Soc. 21 (2008), 615-669
MSC (2000): Primary 37J40, 37J50
DOI: https://doi.org/10.1090/S0894-0347-08-00591-2
Published electronically: March 31, 2008
MathSciNet review: 2393423
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: We study the evolution, under convex Hamiltonian flows on cotangent bundles of compact manifolds, of certain distinguished subsets of the phase space. These subsets are generalizations of Lagrangian graphs, which we call pseudographs. They emerge in a natural way from Fathi's weak KAM theory. By this method, we find various orbits which connect prescribed regions of the phase space. Our study was inspired by works of John Mather. As an application, we obtain the existence of diffusion in a large class of a priori unstable systems and provide a solution to the large gap problem. We hope that our method will have applications to more examples.

RéSUMé. Nous étudions l'évolution, par le flot d'un Hamiltonien convexe sur une variété compacte, de certains ensembles de l'espace des phases. Nous appelons pseudographes ces ensembles, qui sont des généralisations de graphes Lagrangiens apparaissant de manière naturelle dans la théorie KAM faible de Fathi. Par cette méthode, nous trouvons diverses orbites qui joignent des domaines donnés de l'espace des phases. Notre étude s'inspire de travaux de John Mather. Nous obtenons l'existence de diffusion dans une large classe de systèmes à priori instables comme application de cette méthode, qui permet de résoudre le probleme de l'écart entre les tores invariants. Nous espérons que la méthode s'appliquera à d'autres exemples.


References [Enhancements On Off] (What's this?)

  • 1. V. I. ARNOLD, Instability of dynamical systems with several degrees of freedom. Sov. Math. Doklady 5 (1964), 581-585. MR 0163026 (29:329)
  • 2. P. BERNARD, Homoclinic orbits to invariant sets of quasi-integrable exact maps. Ergodic Theory Dynam. Systems 20 (2000), no. 6, 1583-1601. MR 1804946 (2001m:37129)
  • 3. P. BERNARD, Connecting orbits of time dependent Lagrangian systems. Ann. Inst. Fourier 52 (2002), 1533-1568. MR 1935556 (2003m:37088)
  • 4. P. BERNARD, The action spectrum near positive definite invariant tori. Bull. Soc. Math. France 131 (2003), no. 4, 603-616. MR 2044497 (2004m:37111)
  • 5. P. BERNARD, G. CONTRERAS, Generic properties of families of Lagrangian systems. Ann. of Math., to appear.
  • 6. M. BERTI, L. BIASCO, P. BOLLE, Drift in phase space: a new variational mechanism with optimal diffusion time. J. Math. Pures Appl. (9) 82 (2003), no. 6, 613-664. MR 1996776 (2005h:37130)
  • 7. U. BESSI, An approach to Arnold's diffusion through the calculus of variations. Nonlinear Anal. 26 (1996), no. 6, 1115-1135. MR 1375654 (97b:58123)
  • 8. S. BOLOTIN, D. TRESCHEV, Unbounded growth of energy in nonautonomous Hamiltonian systems. Nonlinearity 12 (1999), no. 2, 365-388. MR 1677779 (99m:58086)
  • 9. C.-Q. CHENG, J. YAN, Existence of diffusion orbits in a priori unstable Hamiltonian systems. J. Differential Geom. 67 (2004), no. 3, 457-517. MR 2153027 (2006d:37110)
  • 10. C.-Q. CHENG, J. YAN, Arnold diffusion in Hamiltonian systems: the a priori unstable case, préprint.
  • 11. G. CONTRERAS, G. P. PATERNAIN, Connecting orbits between static classes for generic Lagrangian systems. Topology 41 (2002), no. 4, 645-666. MR 1905833 (2003i:37059)
  • 12. G. CONTRERAS, J. DELGADO, R. ITURRIAGA, Lagrangian flows: the dynamics of globally minimizing orbits. II. Bol. Soc. Brasil. Mat. (N.S.) 28 (1997), no. 2, 155-196. MR 1479500 (98i:58093)
  • 13. A. DELSHAMS, R. DE LA LLAVE, T. SEARA, A geometric mechanism for diffusion in Hamiltonian systems overcoming the large gap problem: announcement of results. Electron. Res. Announc. Amer. Math. Soc. 9 (2003), 125-134 (electronic). MR 2029474 (2004j:37118)
  • 14. W. E, Aubry-Mather theory and periodic solutions of the forced Burgers equation. Commun. Pure Appl. Math. 52 (1999), no. 7, 811-828. MR 1682812 (2000b:37068)
  • 15. A. FATHI, Book in preparation.
  • 16. A. FATHI, Théorème KAM faible et théorie de Mather sur les systèmes lagrangiens. (French) [A weak KAM theorem and Mather's theory of Lagrangian systems]. C. R. Acad. Sci. Paris Ser. I Math. 324 (1997), no. 9, 1043-1046. MR 1451248 (98g:58151)
  • 17. A. FATHI, Solutions KAM faibles conjuguées et barrières de Peierls. (French) [Weakly conjugate KAM solutions and Peierls's barriers]. C. R. Acad. Sci. Paris Ser. I Math. 325 (1997), no. 6, 649-652. MR 1473840 (99b:58209)
  • 18. A. FATHI, Orbites hétéroclines et ensemble de Peierls. (French) [Heteroclinic orbits and the Peierls set]. C. R. Acad. Sci. Paris Ser. I Math. 326 (1998), no. 10, 1213-1216. MR 1650195 (99k:58066)
  • 19. V. KALOSHIN, Geometric proofs of Mather's connecting and accelerating theorems, Topics in dynamics and ergodic theory. London Math. Soc. Lecture Note Ser. 310 (2003), 81-106, Cambridge Univ. Press, Cambridge. MR 2052276 (2005c:37114)
  • 20. Y. KATZNELSON, D. ORNSTEIN, Twist maps and Aubry-Mather sets. Lipa's legacy. Contemp. Math., Amer. Math. Soc., Providence, RI, 211 (1997), 343-357. MR 1476996 (98j:58067)
  • 21. P. LOCHAK, Arnold diffusion; a compendium of remarks and questions, in Hamiltonian systems with three or more degrees of freedom. Kluwer Acad. Publ., Dordrecht, 1999, 168-183. MR 1720892 (2001g:37107)
  • 22. J. P. MARCO, D. SAUZIN, Stability and instability for Gevrey quasi-convex near-integrable Hamiltonian systems. Publ. Math. I. H. E. S. 96 (2002), 199-275 (2003). MR 1986314 (2004m:37112)
  • 23. J. N. MATHER, Action minimizing invariant measures for positive definite Lagrangian systems. Math. Z. 207 (1991), 169-207. MR 1109661 (92m:58048)
  • 24. J. N. MATHER, Variational construction of connecting orbits. Ann. Inst. Fourier 43 (1993), 1349-1368. MR 1275203 (95c:58075)
  • 25. J. N. MATHER, Variational construction of trajectories for time-periodic Lagrangian systems on the two torus, unpublished manuscript.
  • 26. J. N. MATHER, Arnold diffusion I: Announcement of results. J. Math. Sci. 124, no. 5 (2004), 5275-5289. MR 2129140 (2005m:37142)
  • 27. R. MAñé, Lagrangian flows: The dynamics of globally minimizing orbits. Bol. Soc. Bras. Mat. 28 (1997), 141-153. MR 1479499 (98i:58092)
  • 28. K. F. SIBURG, The principle of least action in geometry and dynamics. Lecture Notes in Mathematics, 1844, Springer-Verlag, Berlin, 2004. MR 2076302 (2005m:37151)
  • 29. D. TRESCHEV, Evolution of slow variables in a priori unstable Hamiltonian systems. Nonlinearity 17 (2004), no. 5, 1803-1841. MR 2086152 (2005g:37116)
  • 30. Z. XIA, Arnold diffusion: a variational construction, in Proceedings of the International Congress of Mathematicians, Vol. II (Berlin, 1998), Extra Vol. II, 867-877 (electronic), (1998). MR 1648133 (99g:58112)
  • 31. Z. XIA, Arnold Diffusion and instabilities in Hamiltonian dynamics, preprint.

Similar Articles

Retrieve articles in Journal of the American Mathematical Society with MSC (2000): 37J40, 37J50

Retrieve articles in all journals with MSC (2000): 37J40, 37J50


Additional Information

Patrick Bernard
Affiliation: Université Paris-Dauphine, CEREMADE, UMR CNRS 7534, Place Marechal Lattre Tassigny, 75775 Paris, Cedex 16, France
Email: patrick.bernard@ceremade.dauphine.fr

DOI: https://doi.org/10.1090/S0894-0347-08-00591-2
Keywords: Arnold's diffusion, Mather sets, weak KAM, Hamilton-Jacobi \nobreak equation
Received by editor(s): October 4, 2004
Published electronically: March 31, 2008
Article copyright: © Copyright 2008 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

American Mathematical Society