Bulletin of the American Mathematical Society

Author(s): Vadim Kaloshin; Mark Levi
Journal: Bull. Amer. Math. Soc. 45 (2008), 409-427.
MSC (2000): Primary 70H08
Posted: April 9, 2008
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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.

1.: Arnold, V. Mathematical methods of classical mechanics, Second edition, Graduate Texts in Mathematics, 60, 2nd edition, 1989. MR 997295 (90c:58046)
2.: Arnold, V. Instabilities in dynamical systems with several degrees of freedom, Sov. Math. Dokl. 5 (1964), 581-585.
3.: Arnold, V. I.; Kozlov, V. V.; Neishtadt, A. I. Mathematical aspects of classical and celestial mechanics. 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. Springer-Verlag, Berlin, 1997. MR 1656199 (2000b:37054)
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.: Bessi, U. An approach to Arnold's diffusion through the calculus of vartiations, Nonlinear Anal. 26(6) (1996), 1115-1135. MR 1375654 (97b:58123)
7.: Bessi, U.; Chierchia, L.; Valdinoci, E. Upper bounds on Arnold diffusion times via Mather theory, J. Math. Pures Appl. 80(1) (2001) 105-129. MR 1810511 (2002a:37093)
8.: Berti, M.; Bolle, Ph. A functional analysis approach to Arnold diffusion, Ann. Inst. H. Poincare 19(4) (2002), 395-450. MR 1912262 (2003g:37105)
9.: Bourgain J.; Kaloshin V. On diffusion in high-dimensional Hamiltonian systems. J. Funct. Anal. 229(1) (2005), 1-61. MR 2180073
10.: Cheng, C.-Q.; Yan, J. Existence of diffusion orbits in a priori unstable Hamiltonian systems. J. Differential Geom. 67(3) (2004), 457-518. MR 2153027 (2006d:37110)
11.: Cheng, C.-Q.; Yan J. Arnold diffusion in Hamiltonian systems: the a priori unstable case, preprint.
12.: Contreras, G.; Iturriaga, R. Global Minimizers of Autonomous Lagrangians. 22 Coloquio Brasileiro de Matematica, IMPA, Rio de Janeiro, 1999. MR 1720372 (2001j:37113)
13.: Delshams, A.; de la Llave, R.; Seara, T. 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 pp. MR 2184276
14.: Fathi, A. The weak KAM theorem in Lagrangian dynamics, Cambridge Studies in Advanced Mathematics, vol. 88, Cambridge Univesity Press, 2003.
15.: Fenichel, N. Persistence and smoothness of invariant manifolds for flows, Indiana Univ. Math. J. 21 (1971), 193-226. MR 0287106 (44:4313)
16.: Hirsch, M.; Pugh, C.; Shub, M. Invariant manifolds, Lect. Notes in Math, vol. 583, Springer, 1977. MR 0501173 (58:18595)
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.: Marco, J.-P.; Sauzin, D. Stability and instability for Gevrey quasi-convex near-integrable Hamiltonian systems. Publ. Math. Inst. Hautes Etudes Sci. No. 96 (2002), 199-275 (2003). MR 1986314 (2004m:37112)
20.: Mather, J. Action minimizing invariant measures for positive definite Lagrangians, Math. Z. 207 (1991), 169-207. MR 1109661 (92m:58048)
21.: Mather, J. Variational construction of connecting orbits, Ann. Inst. Fourier 43 (1993), 1349-1386. MR 1275203 (95c:58075)
22.: Mather, J. Modulus of continuity of Peierls' barrier, in NATO ASI Series C: vol. 209, Periodic Solutions of Hamiltonian Systems and Related Topics, edited by P. Rabinowitz, 1987, 177-202. MR 920622 (89c:58109)
23.: Mather, J. Arnold diffusion. I: Announcement of results, J. of Math. Sciences 124(5) (2004), 5275-5289. MR 2129140 (2005m:37142)
24.: Mather, J. Total disconnectedness of the quotient Aubry set in low dimensions, Comm. Pure Appl. Math. 56(8) (2003), 1178-1183. MR 1989233 (2004c:37152)
25.: Mather, J. Graduate Class 2001-2002, Princeton, 2002.
26.: Mather, J. Arnold diffusion. II, preprint, 2006, 160pp.
27.: Nekhoroshev, N. An exponential estimate of the time of stability of nearly integrable Hamiltonian systems, Uspekhi Math. Nauk 32(1) (1977), 5-66. MR 0501140 (58:18570)
28.: Siburg, K. The Principle of Least Action in Geometry and Dynamics, Lecture Notes in Mathematics, 1844, Springer 2004. MR 2076302 (2005m:37151)
29.: Treschev, D. Multidimensional symplectic separatrix maps, J. Nonlinear Sci. 12(1) (2002), 27-58. MR 1888569 (2002m:37089)
30.: Treschev, D. Evolution of slow variables in a priori unstable Hamiltonian systems. Nonlinearity 17(5) (2004), 1803-1841. MR 2086152 (2005g:37116)
31.: Xia, J. Arnold diffusion: a variational construction. Proc of the ICM, vol. II (Berlin, 1998), 867-877, 1998. MR 1648133 (99g:58112)

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

Vadim Kaloshin
Affiliation: Mathematics 253-37, California Institute of Technology, Pasadena, California 91125
Email: kaloshin@math.psu.edu

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

DOI: 10.1090/S0273-0979-08-01211-1
PII: S 0273-0979(08)01211-1
Received by editor(s): March 3, 2007,
Received by editor(s) in revised form: September 17, 2007
Posted: 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
Copyright of article: Copyright 2008, American Mathematical Society

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