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An example of Arnold diffusion for near-integrable Hamiltonians. (English) Zbl 1141.70009
Summary: Using the ideas of U. Bessi [Nonlinear Anal., Theory Methods Appl. 26, No. 6, 1115–1135 (1996; Zbl 0867.70013)] and J. Mather [J. Graduate Class 2001–2002. Princeton (2002)], 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.

##### MSC:
 70H08 Nearly integrable Hamiltonian systems, KAM theory 78A05 Geometric optics
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##### References:
 [1] V. I. Arnol$$^{\prime}$$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. [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)]. · Zbl 0885.70001 [4] Bernard, P. Dynamics of pseudographs in convex Hamiltonian systems, to appear in the Journal of the AMS. · Zbl 1213.37089 [5] Bernard, P.; Contreras, G. A generic property of families of Lagrangian systems, to appear in the Annals of Mathematics. · Zbl 1175.37067 [6] Ugo Bessi, An approach to Arnol$$^{\prime}$$d’s diffusion through the calculus of variations, Nonlinear Anal. 26 (1996), no. 6, 1115 – 1135. · Zbl 0867.70013 [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. · Zbl 0986.37052 [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). · Zbl 1087.37048 [9] Jean Bourgain and Vadim Kaloshin, On diffusion in high-dimensional Hamiltonian systems, J. Funct. Anal. 229 (2005), no. 1, 1 – 61. · Zbl 1080.37062 [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. · Zbl 1098.37055 [11] Cheng, C.-Q.; Yan J. Arnold diffusion in Hamiltonian systems: the a priori unstable case, preprint. · Zbl 1179.37081 [12] Gonzalo Contreras and Renato Iturriaga, Global minimizers of autonomous Lagrangians, 22^{\?} Colóquio Brasileiro de Matemática. [22nd Brazilian Mathematics Colloquium], Instituto de Matemática Pura e Aplicada (IMPA), Rio de Janeiro, 1999. · Zbl 0957.37065 [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. · Zbl 1090.37044 [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. · Zbl 0246.58015 [16] M. W. Hirsch, C. C. Pugh, and M. Shub, Invariant manifolds, Lecture Notes in Mathematics, Vol. 583, Springer-Verlag, Berlin-New York, 1977. · Zbl 0355.58009 [17] Kaloshin, V.; Levi, M. Geometry of Arnold diffusion, to appear in SIAM Review. · Zbl 1152.37024 [18] Levi, M. Shadowing property of geodesics in Hedlund’s metric, Ergo. Th. & Dynam. Syst. 17 (1997), 187-203. · Zbl 0879.58062 [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). · Zbl 1086.37031 [20] John N. Mather, Action minimizing invariant measures for positive definite Lagrangian systems, Math. Z. 207 (1991), no. 2, 169 – 207. · Zbl 0696.58027 [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). · Zbl 0803.58019 [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. · Zbl 0658.58013 [23] Dzh. N. Mèzer, Arnol$$^{\prime}$$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. · Zbl 1069.37044 [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. · Zbl 1046.37039 [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). [28] Karl Friedrich Siburg, The principle of least action in geometry and dynamics, Lecture Notes in Mathematics, vol. 1844, Springer-Verlag, Berlin, 2004. · Zbl 1060.37048 [29] D. Treschev, Multidimensional symplectic separatrix maps, J. Nonlinear Sci. 12 (2002), no. 1, 27 – 58. · Zbl 1022.37041 [30] D. Treschev, Evolution of slow variables in a priori unstable Hamiltonian systems, Nonlinearity 17 (2004), no. 5, 1803 – 1841. · Zbl 1075.37019 [31] Zhihong Xia, Arnold diffusion: a variational construction, Proceedings of the International Congress of Mathematicians, Vol. II (Berlin, 1998), 1998, pp. 867 – 877. · Zbl 0910.58015
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