Remote Access Bulletin of the American Mathematical Society

Bulletin of the American Mathematical Society

ISSN 1088-9485(online) ISSN 0273-0979(print)



An example of Arnold diffusion for near-integrable Hamiltonians

Authors: Vadim Kaloshin and Mark Levi
Journal: Bull. Amer. Math. Soc. 45 (2008), 409-427
MSC (2000): Primary 70H08
Published electronically: April 9, 2008
MathSciNet review: 2402948
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: In this paper, using the ideas of Bessi and Mather, we present a simple mechanical system exhibiting Arnold diffusion. This system of a particle in a small periodic potential can be also interpreted as ray propagation in a periodic optical medium with a near-constant index of refraction. Arnold diffusion in this context manifests itself as an arbitrary finite change of direction for nearly constant index of refraction.

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

  • 1. V. I. Arnol′d, Mathematical methods of classical mechanics, 2nd ed., Graduate Texts in Mathematics, vol. 60, Springer-Verlag, New York, 1989. Translated from the Russian by K. Vogtmann and A. Weinstein. MR 997295
  • 2. Arnold, V. Instabilities in dynamical systems with several degrees of freedom, Sov. Math. Dokl. 5 (1964), 581-585.
  • 3. V. I. Arnold, V. V. Kozlov, and A. I. Neishtadt, Mathematical aspects of classical and celestial mechanics, Springer-Verlag, Berlin, 1997. Translated from the 1985 Russian original by A. Iacob; Reprint of the original English edition from the series Encyclopaedia of Mathematical Sciences [Dynamical systems. III, Encyclopaedia Math. Sci., 3, Springer, Berlin, 1993; MR1292465 (95d:58043a)]. MR 1656199
  • 4. Bernard, P. Dynamics of pseudographs in convex Hamiltonian systems, to appear in the Journal of the AMS.
  • 5. Bernard, P.; Contreras, G. A generic property of families of Lagrangian systems, to appear in the Annals of Mathematics.
  • 6. Ugo Bessi, An approach to Arnol′d’s diffusion through the calculus of variations, Nonlinear Anal. 26 (1996), no. 6, 1115–1135. MR 1375654, 10.1016/0362-546X(94)00270-R
  • 7. Ugo Bessi, Luigi Chierchia, and Enrico Valdinoci, Upper bounds on Arnold diffusion times via Mather theory, J. Math. Pures Appl. (9) 80 (2001), no. 1, 105–129. MR 1810511, 10.1016/S0021-7824(00)01188-0
  • 8. Massimiliano Berti and Philippe Bolle, A functional analysis approach to Arnold diffusion, Ann. Inst. H. Poincaré Anal. Non Linéaire 19 (2002), no. 4, 395–450 (English, with English and French summaries). MR 1912262, 10.1016/S0294-1449(01)00084-1
  • 9. Jean Bourgain and Vadim Kaloshin, On diffusion in high-dimensional Hamiltonian systems, J. Funct. Anal. 229 (2005), no. 1, 1–61. MR 2180073, 10.1016/j.jfa.2004.09.006
  • 10. Chong-Qing Cheng and Jun Yan, Existence of diffusion orbits in a priori unstable Hamiltonian systems, J. Differential Geom. 67 (2004), no. 3, 457–517. MR 2153027
  • 11. Cheng, C.-Q.; Yan J. Arnold diffusion in Hamiltonian systems: the a priori unstable case, preprint.
  • 12. Gonzalo Contreras and Renato Iturriaga, Global minimizers of autonomous Lagrangians, 22^{𝑜} Colóquio Brasileiro de Matemática. [22nd Brazilian Mathematics Colloquium], Instituto de Matemática Pura e Aplicada (IMPA), Rio de Janeiro, 1999. MR 1720372
  • 13. Amadeu Delshams, Rafael de la Llave, and Tere M. Seara, A geometric mechanism for diffusion in Hamiltonian systems overcoming the large gap problem: heuristics and rigorous verification on a model, Mem. Amer. Math. Soc. 179 (2006), no. 844, viii+141. MR 2184276, 10.1090/memo/0844
  • 14. Fathi, A. The weak KAM theorem in Lagrangian dynamics, Cambridge Studies in Advanced Mathematics, vol. 88, Cambridge Univesity Press, 2003.
  • 15. Neil Fenichel, Persistence and smoothness of invariant manifolds for flows, Indiana Univ. Math. J. 21 (1971/1972), 193–226. MR 0287106
  • 16. M. W. Hirsch, C. C. Pugh, and M. Shub, Invariant manifolds, Lecture Notes in Mathematics, Vol. 583, Springer-Verlag, Berlin-New York, 1977. MR 0501173
  • 17. Kaloshin, V.; Levi, M. Geometry of Arnold diffusion, to appear in SIAM Review.
  • 18. Levi, M. Shadowing property of geodesics in Hedlund's metric, Ergo. Th. & Dynam. Syst. 17 (1997), 187-203.
  • 19. Jean-Pierre Marco and David Sauzin, Stability and instability for Gevrey quasi-convex near-integrable Hamiltonian systems, Publ. Math. Inst. Hautes Études Sci. 96 (2002), 199–275 (2003). MR 1986314, 10.1007/s10240-003-0011-5
  • 20. John N. Mather, Action minimizing invariant measures for positive definite Lagrangian systems, Math. Z. 207 (1991), no. 2, 169–207. MR 1109661, 10.1007/BF02571383
  • 21. John N. Mather, Variational construction of connecting orbits, Ann. Inst. Fourier (Grenoble) 43 (1993), no. 5, 1349–1386 (English, with English and French summaries). MR 1275203
  • 22. John N. Mather, Modulus of continuity for Peierls’s barrier, Periodic solutions of Hamiltonian systems and related topics (Il Ciocco, 1986) NATO Adv. Sci. Inst. Ser. C Math. Phys. Sci., vol. 209, Reidel, Dordrecht, 1987, pp. 177–202. MR 920622
  • 23. Dzh. N. Mèzer, Arnol′d diffusion. I. Announcement of results, Sovrem. Mat. Fundam. Napravl. 2 (2003), 116–130 (Russian, with Russian summary); English transl., J. Math. Sci. (N. Y.) 124 (2004), no. 5, 5275–5289. MR 2129140, 10.1023/B:JOTH.0000047353.78307.09
  • 24. John N. Mather, Total disconnectedness of the quotient Aubry set in low dimensions, Comm. Pure Appl. Math. 56 (2003), no. 8, 1178–1183. Dedicated to the memory of Jürgen K. Moser. MR 1989233, 10.1002/cpa.10091
  • 25. Mather, J. Graduate Class 2001-2002, Princeton, 2002.
  • 26. Mather, J. Arnold diffusion. II, preprint, 2006, 160pp.
  • 27. N. N. Nehorošev, An exponential estimate of the time of stability of nearly integrable Hamiltonian systems, Uspehi Mat. Nauk 32 (1977), no. 6(198), 5–66, 287 (Russian). MR 0501140
  • 28. Karl Friedrich Siburg, The principle of least action in geometry and dynamics, Lecture Notes in Mathematics, vol. 1844, Springer-Verlag, Berlin, 2004. MR 2076302
  • 29. D. Treschev, Multidimensional symplectic separatrix maps, J. Nonlinear Sci. 12 (2002), no. 1, 27–58. MR 1888569, 10.1007/s00332-001-0460-2
  • 30. D. Treschev, Evolution of slow variables in a priori unstable Hamiltonian systems, Nonlinearity 17 (2004), no. 5, 1803–1841. MR 2086152, 10.1088/0951-7715/17/5/014
  • 31. Zhihong Xia, Arnold diffusion: a variational construction, Proceedings of the International Congress of Mathematicians, Vol. II (Berlin, 1998), 1998, pp. 867–877 (electronic). MR 1648133

Similar Articles

Retrieve articles in Bulletin of the American Mathematical Society with MSC (2000): 70H08

Retrieve articles in all journals with MSC (2000): 70H08

Additional Information

Vadim Kaloshin
Affiliation: Mathematics 253-37, California Institute of Technology, Pasadena, California 91125

Mark Levi
Affiliation: Mathematics 253-37, California Institute of Technology, Pasadena, California 91125

Received by editor(s): March 3, 2007
Received by editor(s) in revised form: September 17, 2007
Published electronically: April 9, 2008
Additional Notes: The first author was partially supported by the Sloan Foundation and NSF grants, DMS-0701271
The second author was partially supported by NSF grant DMS-0605878
Article copyright: © Copyright 2008 American Mathematical Society