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Periodic orbits for Hamiltonian systems in cotangent bundles


Author: Christophe Golé
Journal: Trans. Amer. Math. Soc. 343 (1994), 327-347
MSC: Primary 58E05; Secondary 34C25, 58F05, 58F22
MathSciNet review: 1232186
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Abstract: We prove the existence of at least $ \operatorname{cl}(M)$ periodic orbits for certain time-dependent Hamiltonian systems on the cotangent bundle of an arbitrary compact manifold M. These Hamiltonians are not necessarily convex but they satisfy a certain boundary condition given by a Riemannian metric on M. We discretize the variational problem by decomposing the time-1 map into a product of "symplectic twist maps". A second theorem deals with homotopically non-trivial orbits of negative curvature.


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  • [AM] Ralph Abraham and Jerrold E. Marsden, Foundations of mechanics, Benjamin/Cummings Publishing Co., Inc., Advanced Book Program, Reading, Mass., 1978. Second edition, revised and enlarged; With the assistance of Tudor Raţiu and Richard Cushman. MR 515141
  • [Ar1] Vladimir Arnol′d, Sur une propriété topologique des applications globalement canoniques de la mécanique classique, C. R. Acad. Sci. Paris 261 (1965), 3719–3722 (French). MR 0193645
  • [Ar2] V. I. Arnold, Mathematical methods of classical mechanics, Springer-Verlag, New York-Heidelberg, 1978. Translated from the Russian by K. Vogtmann and A. Weinstein; Graduate Texts in Mathematics, 60. MR 0690288
  • [AL] S. Aubry and P. Y. Le Daeron, The discrete Frenkel-Kontorova model and its extensions. I. Exact results for the ground-states, Phys. D 8 (1983), no. 3, 381–422. MR 719634, 10.1016/0167-2789(83)90233-6
  • [BG] A. Banyaga and C. Golé, A remark on a conjecture of Arnold: linked spheres and fixed points, Proc. Conf. on Hamiltonian Systems and Celestial Mechanics (Guanajuato) (1993).
  • [BK] David Bernstein and Anatole Katok, Birkhoff periodic orbits for small perturbations of completely integrable Hamiltonian systems with convex Hamiltonians, Invent. Math. 88 (1987), no. 2, 225–241. MR 880950, 10.1007/BF01388907
  • [BP] M. Baily and L. Polterovitch, Hamiltonian diffeomorphisms and Lagrangian distributions, preprint, Tel Aviv University, 1991.
  • [Ch1] M. Chaperon, Quelques questions de géométrie symplectique, Séminaire. Bourbaki, no. 610, 1982/83.
  • [Ch2] Marc Chaperon, Une idée du type “géodésiques brisées” pour les systèmes hamiltoniens, C. R. Acad. Sci. Paris Sér. I Math. 298 (1984), no. 13, 293–296 (French, with English summary). MR 765426
  • [Co] Charles Conley, Isolated invariant sets and the Morse index, CBMS Regional Conference Series in Mathematics, vol. 38, American Mathematical Society, Providence, R.I., 1978. MR 511133
  • [CZ1] C. C. Conley and E. Zehnder, The Birkhoff-Lewis fixed point theorem and a conjecture of V. I. Arnol′d, Invent. Math. 73 (1983), no. 1, 33–49. MR 707347, 10.1007/BF01393824
  • [CZ2] Charles Conley and Eduard Zehnder, Morse-type index theory for flows and periodic solutions for Hamiltonian equations, Comm. Pure Appl. Math. 37 (1984), no. 2, 207–253. MR 733717, 10.1002/cpa.3160370204
  • [D] Raphaël Douady, Stabilité ou instabilité des points fixes elliptiques, Ann. Sci. École Norm. Sup. (4) 21 (1988), no. 1, 1–46 (French). MR 944100
  • [DNF] B. Doubrovine, S. Novikov, and A. Fomenko, Géométrie contemporaine, vol. 3, "Mir", Moscow, 1987 (see also English translation in Springer-Verlag).
  • [F1] Andreas Floer, A refinement of the Conley index and an application to the stability of hyperbolic invariant sets, Ergodic Theory Dynam. Systems 7 (1987), no. 1, 93–103. MR 886372, 10.1017/S0143385700003825
  • [F2] Andreas Floer, Morse theory for Lagrangian intersections, J. Differential Geom. 28 (1988), no. 3, 513–547. MR 965228
  • [GHL] S. Gallot, D. Hulin, and J. Lafontaine, Riemannian geometry, Universitext, Springer-Verlag, Berlin, 1987. MR 909697
  • [G1] C. Golé, Periodic points for monotone symplectomorphisms of $ {\mathbb{T}^n} \times {\mathbb{R}^n}$, Ph.D. thesis, Boston University, 1989.
  • [G2] Christophe Golé, Ghost circles for twist maps, J. Differential Equations 97 (1992), no. 1, 140–173. MR 1161316, 10.1016/0022-0396(92)90088-5
  • [GH] Christophe Golé and Glen R. Hall, Poincaré’s proof of Poincaré’s last geometric theorem, Twist mappings and their applications, IMA Vol. Math. Appl., vol. 44, Springer, New York, 1992, pp. 135–151. MR 1219354, 10.1007/978-1-4613-9257-6_8
  • [Gr] M. J. Greenberg, Lectures on algebraic topology, Math. Lecture Note Ser., 5th printing, 1977.
  • [H] Michael-R. Herman, Existence et non existence de tores invariants par des difféomorphismes symplectiques, Séminaire sur les Équations aux Dérivées Partielles 1987–1988, École Polytech., Palaiseau, 1988, pp. Exp. No. XIV, 24 (French). MR 1018186
  • [J] F. W. Josellis, Global periodic orbits for Hamiltonian systems on $ {{\mathbf{T}}^n} \times {{\mathbf{R}}^n}$, Ph.D. thesis, no. 9518, ETH Zürich, 1991.
  • [Ka] A. Katok, Some remarks of Birkhoff and Mather twist map theorems, Ergodic Theory Dynamical Systems 2 (1982), no. 2, 185–194 (1983). MR 693974
  • [K1] Wilhelm Klingenberg, Riemannian geometry, de Gruyter Studies in Mathematics, vol. 1, Walter de Gruyter & Co., Berlin-New York, 1982. MR 666697
  • [K-M] Hyung-tae Kook and James D. Meiss, Periodic orbits for reversible, symplectic mappings, Phys. D 35 (1989), no. 1-2, 65–86. MR 1004186, 10.1016/0167-2789(89)90096-1
  • [L] P. LeCalvez, Existence d'orbits de Birkhoff généralisées pour les difféomorphismes conservatifs de l'anneau, preprint, Univ. Paris-Sud, Orsay, 1989.
  • [McD] Dusa McDuff, Elliptic methods in symplectic geometry, Bull. Amer. Math. Soc. (N.S.) 23 (1990), no. 2, 311–358. MR 1039425, 10.1090/S0273-0979-1990-15928-2
  • [MM] R. S. MacKay and J. D. Meiss, Linear stability of periodic orbits in Lagrangian systems, Phys. Lett. A 98 (1983), no. 3, 92–94. MR 721605, 10.1016/0375-9601(83)90735-1
  • [MMS] R. S. MacKay, J. D. Meiss, and J. Stark, Converse KAM theory for symplectic twist maps, Nonlinearity 2 (1989), no. 4, 555–570. MR 1020442
  • [Ma] 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
  • [Mi] J. Milnor, Morse theory, Based on lecture notes by M. Spivak and R. Wells. Annals of Mathematics Studies, No. 51, Princeton University Press, Princeton, N.J., 1963. MR 0163331
  • [Mo1] J. Moser, Proof of a generalized form of a fixed point theorem due to G. D. Birkhoff, Geometry and topology (Proc. III Latin Amer. School of Math., Inst. Mat. Pura Aplicada CNPq, Rio de Janeiro, 1976) Springer, Berlin, 1977, pp. 464–494. Lecture Notes in Math., Vol. 597. MR 0494305
  • [Mo2] Jürgen Moser, Monotone twist mappings and the calculus of variations, Ergodic Theory Dynam. Systems 6 (1986), no. 3, 401–413. MR 863203, 10.1017/S0143385700003588
  • [S] Jean-Claude Sikorav, Sur les immersions lagrangiennes dans un fibré cotangent admettant une phase génératrice globale, C. R. Acad. Sci. Paris Sér. I Math. 302 (1986), no. 3, 119–122 (French, with English summary). MR 830282
  • [V] Claude Viterbo, Intersection de sous-variétés lagrangiennes, fonctionnelles d’action et indice des systèmes hamiltoniens, Bull. Soc. Math. France 115 (1987), no. 3, 361–390 (French, with English summary). MR 926533
  • [V2] -, Symplectic topology as the geometry of generaing functions, preprint, Univ. Paris-Dauphine, 1991.

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DOI: https://doi.org/10.1090/S0002-9947-1994-1232186-5
Article copyright: © Copyright 1994 American Mathematical Society