## An example of Arnold diffusion for near-integrable Hamiltonians

HTML articles powered by AMS MathViewer

- by Vadim Kaloshin and Mark Levi PDF
- Bull. Amer. Math. Soc.
**45**(2008), 409-427 Request permission

## 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

- 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**, DOI 10.1007/978-1-4757-2063-1
A2 Arnold, V. Instabilities in dynamical systems with several degrees of freedom, Sov. Math. Dokl. - 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**
B Bernard, P. Dynamics of pseudographs in convex Hamiltonian systems, to appear in the Journal of the AMS.
BC Bernard, P.; Contreras, G. A generic property of families of Lagrangian systems, to appear in the Annals of Mathematics.
- Ugo Bessi,
*An approach to Arnol′d’s diffusion through the calculus of variations*, Nonlinear Anal.**26**(1996), no. 6, 1115–1135. MR**1375654**, DOI 10.1016/0362-546X(94)00270-R - 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**, DOI 10.1016/S0021-7824(00)01188-0 - Massimiliano Berti and Philippe Bolle,
*A functional analysis approach to Arnold diffusion*, Ann. Inst. H. Poincaré C Anal. Non Linéaire**19**(2002), no. 4, 395–450 (English, with English and French summaries). MR**1912262**, DOI 10.1016/S0294-1449(01)00084-1 - Jean Bourgain and Vadim Kaloshin,
*On diffusion in high-dimensional Hamiltonian systems*, J. Funct. Anal.**229**(2005), no. 1, 1–61. MR**2180073**, DOI 10.1016/j.jfa.2004.09.006 - 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**
CY2 Cheng, C.-Q.; Yan J. Arnold diffusion in Hamiltonian systems: the a priori unstable case, preprint.
- Gonzalo Contreras and Renato Iturriaga,
*Global minimizers of autonomous Lagrangians*, 22$^\textrm {o}$ Colóquio Brasileiro de Matemática. [22nd Brazilian Mathematics Colloquium], Instituto de Matemática Pura e Aplicada (IMPA), Rio de Janeiro, 1999. MR**1720372** - 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**, DOI 10.1090/memo/0844
Fa Fathi, A. The weak KAM theorem in Lagrangian dynamics, Cambridge Studies in Advanced Mathematics, vol. 88, Cambridge Univesity Press, 2003.
- Neil Fenichel,
*Persistence and smoothness of invariant manifolds for flows*, Indiana Univ. Math. J.**21**(1971/72), 193–226. MR**287106**, DOI 10.1512/iumj.1971.21.21017 - 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**
KL Kaloshin, V.; Levi, M. Geometry of Arnold diffusion, to appear in SIAM Review.
Le Levi, M. Shadowing property of geodesics in Hedlund’s metric, Ergo. Th. & Dynam. Syst. - 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**, DOI 10.1007/s10240-003-0011-5 - John N. Mather,
*Action minimizing invariant measures for positive definite Lagrangian systems*, Math. Z.**207**(1991), no. 2, 169–207. MR**1109661**, DOI 10.1007/BF02571383 - 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** - 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** - 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**, DOI 10.1023/B:JOTH.0000047353.78307.09 - 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**, DOI 10.1002/cpa.10091
Ma6 Mather, J. Graduate Class 2001-2002, Princeton, 2002.
Ma7 Mather, J. Arnold diffusion. II, preprint, 2006, 160pp.
- 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** - Karl Friedrich Siburg,
*The principle of least action in geometry and dynamics*, Lecture Notes in Mathematics, vol. 1844, Springer-Verlag, Berlin, 2004. MR**2076302**, DOI 10.1007/b97327 - D. Treschev,
*Multidimensional symplectic separatrix maps*, J. Nonlinear Sci.**12**(2002), no. 1, 27–58. MR**1888569**, DOI 10.1007/s00332-001-0460-2 - D. Treschev,
*Evolution of slow variables in a priori unstable Hamiltonian systems*, Nonlinearity**17**(2004), no. 5, 1803–1841. MR**2086152**, DOI 10.1088/0951-7715/17/5/014 - Zhihong Xia,
*Arnold diffusion: a variational construction*, Proceedings of the International Congress of Mathematicians, Vol. II (Berlin, 1998), 1998, pp. 867–877. MR**1648133**

**5**(1964), 581–585.

**17**(1997), 187–203.

## Additional Information

**Vadim Kaloshin**- Affiliation: Mathematics 253-37, California Institute of Technology, Pasadena, California 91125
- MR Author ID: 624885
- Email: kaloshin@math.psu.edu
**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 - © Copyright 2008 American Mathematical Society
- Journal: Bull. Amer. Math. Soc.
**45**(2008), 409-427 - MSC (2000): Primary 70H08
- DOI: https://doi.org/10.1090/S0273-0979-08-01211-1
- MathSciNet review: 2402948