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Hamiltonian Systems

  • Gerd Rudolph
  • Matthias Schmidt
Part of the Theoretical and Mathematical Physics book series (TMP)

Abstract

We start with an introductory part, including the Legendre transformation, a brief discussion of linear nonholonomic systems and a presentation of three important classes of examples: the geodesic flow, Hamiltonian systems on Lie group manifolds and Hamiltonian systems on coadjoint orbits. We finish this part by showing how to deal with time-dependent Hamiltonian systems. Next, we investigate the structure of regular energy surfaces and discuss the problem of the existence of periodic integral curves for autonomous Hamiltonian systems. This leads us to the famous Weinstein conjecture and to symplectic capacities. Thereafter, we investigate the behaviour of a Hamiltonian system near a critical integral curve. We show that periodic integral curves generically come in orbit cylinders and prove the Lyapunov Center Theorem. We derive the Birkhoff normal form both for symplectomorphisms near an elliptic fixed point and for the Hamiltonian of a system near an equilibrium. The normal form of the Hamiltonian induces a foliation of the phase space into invariant tori so that, in the normal form approximation, the theory becomes integrable. Moreover, we prove the Birkhoff-Lewis Theorem, which states that under a certain nonresonance condition, near a periodic integral curve there exist infinitely many periodic points which lie on the same energy surface. Thereafter, we discuss some aspects of stability, with the main emphasis on systems with two degrees of freedom. In the final two sections we study time-dependent Hamiltonian systems. This includes a discussion of the stability problem of time-periodic systems with emphasis on parametric resonance and an introduction to the famous Arnold conjecture about the number of fixed points of Hamiltonian symplectomorphisms.

Keywords

Hamiltonian System Integral Curve Integral Curf Coadjoint Orbit Hamiltonian Vector Field 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

References

  1. 1.
    Abraham, R., Marsden, J.E.: Foundations of Mechanics. Benjamin-Cummings, Reading (1978) zbMATHGoogle Scholar
  2. 3.
    Abramowitz, M., Stegun, I.A.: Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables. National Bureau of Standards Applied Mathematics Series, vol. 55 (1964) zbMATHGoogle Scholar
  3. 11.
    Arnold, V.I.: Proof of A.N. Kolmogorov’s theorem on the preservation of quasi-periodic motions under small perturbations of the Hamiltonian. Russ. Math. Surv. 18, 9–36 (1963) CrossRefGoogle Scholar
  4. 12.
    Arnold, V.I.: Small divisor problems in classical and celestial mechanics. Russ. Math. Surv. 18, 85–192 (1963) CrossRefGoogle Scholar
  5. 18.
    Arnold, V.I.: Mathematische Methoden der klassischen Mechanik. Birkhäuser, Basel (1988) Google Scholar
  6. 22.
    Arnold, V.I.: Symplectic geometry and topology. J. Math. Phys. 41(6), 3307–3343 (2000) MathSciNetADSzbMATHCrossRefGoogle Scholar
  7. 26.
    Arnold, V.I., Kozlov, V.V., Neishtadt, A.I.: Mathematical aspects of classical and celestial mechanics. In: Arnold, V.I. (ed.) Dynamical Systems III. Springer, Berlin (1988) Google Scholar
  8. 35.
    Bates, L., Śniatycki, J.: Nonholonomic reduction. Rep. Math. Phys. 32(1), 99–115 (1993) MathSciNetADSzbMATHCrossRefGoogle Scholar
  9. 45.
    Birkhoff, G.D.: Proof of Poincaré’s geometric theorem. Trans. Am. Math. Soc. 14, 14–22 (1913) MathSciNetzbMATHGoogle Scholar
  10. 46.
    Birkhoff, G.D.: An extension of Poincaré’s last geometric theorem. Acta Math. 47, 297–311 (1925) MathSciNetCrossRefGoogle Scholar
  11. 47.
    Birkhoff, G.D.: Dynamical Systems. American Mathematical Society Colloquium Publications, vol. IX (1927) zbMATHGoogle Scholar
  12. 48.
    Birkhoff, G.D., Lewis, D.C.: On the periodic motions near a given periodic motion of a dynamical system. Ann. Mat. Pura Appl. 12, 117–133 (1933) MathSciNetzbMATHCrossRefGoogle Scholar
  13. 63.
    Chierchia, L.: Kolmogorov-Arnold-Moser (KAM) theory. In: Meyers, R.A. (editor-in-chief) Encyclopedia of Complexity and Systems Science, pp. 5064–5091. Springer, Berlin (2009) Google Scholar
  14. 66.
    Conley, C., Zehnder, E.: The Birkhoff-Lewis fixed point theorem and a conjecture of V.I. Arnold. Invent. Math. 73, 33–49 (1983) MathSciNetADSzbMATHCrossRefGoogle Scholar
  15. 75.
    Douady, R.: Une démonstration directe de l’équivalence des théorèmes de tores invariants pour difféomorphismes et champs des vecteurs. C. R. Acad. Sci. Paris Sér. I Math. 295, 201–204 (1982) MathSciNetzbMATHGoogle Scholar
  16. 76.
    Dubrovin, B.A., Novikov, S.P., Fomenko, A.T.: Modern Geometry—Methods and Applications. Springer, Berlin (1992) zbMATHCrossRefGoogle Scholar
  17. 84.
    Ekeland, I., Hofer, H.: Symplectic topology and Hamiltonian dynamics. Math. Z. 200, 355–378 (1990) MathSciNetCrossRefGoogle Scholar
  18. 85.
    Eliashberg, Y.: Estimates on the number of fixed points of area preserving transformations. Syktyvkar University preprint (1979) Google Scholar
  19. 88.
    Fadell, E., Rabinowitz, P.: Generalized cohomological index theories for the group actions with an application to bifurcation questions for Hamiltonian systems. Invent. Math. 45, 139–174 (1978) MathSciNetADSzbMATHCrossRefGoogle Scholar
  20. 93.
    Floer, A.: The unregularized gradient flow of the symplectic action. Commun. Pure Appl. Math. 41, 775–813 (1988) MathSciNetzbMATHCrossRefGoogle Scholar
  21. 94.
    Floer, A.: Symplectic fixed points and holomorphic spheres. Commun. Math. Phys. 120, 513–547 (1989) MathSciNetCrossRefGoogle Scholar
  22. 100.
    Fukaya, K., Ono, K.: Arnold conjecture and Gromov-Witten invariants. Topology 38, 933–1048 (1999) MathSciNetzbMATHCrossRefGoogle Scholar
  23. 104.
    Ginzburg, V.L.: Some remarks on symplectic actions of compact groups. Math. Z. 210, 625–640 (1992) MathSciNetzbMATHCrossRefGoogle Scholar
  24. 105.
    Ginzburg, V.L.: An embedding S 2n−1→ℝ2n, 2n−1≥7, whose Hamiltonian flow has no periodic trajectories. Int. Math. Res. Not. 2, 83–98 (1995) CrossRefGoogle Scholar
  25. 106.
    Ginzburg, V.L.: The Weinstein conjecture and the theorems of nearby and almost existence. In: Marsden, J.E., Ratiu, T.S. (eds.) The Breadth of Symplectic and Poisson Geometry. Festschrift in Honor of Alan Weinstein, pp. 139–172. Birkhäuser, Basel (2005) Google Scholar
  26. 135.
    Hofer, H.: Pseudoholomorphic curves in symplectizations with applications to the Weinstein conjecture in dimension three. Invent. Math. 114, 515–563 (1993) MathSciNetADSzbMATHCrossRefGoogle Scholar
  27. 136.
    Hofer, H.: Dynamics, topology and holomorphic curves. In: Proceedings of the International Congress of Mathematicians, Berlin, 1998. Doc. Math., extra vol. I pp. 255–280 (1998) Google Scholar
  28. 137.
    Hofer, H., Salamon, D.: Floer homology and Novikov rings. In: The Floer Memorial Volume. Progress in Mathematics, vol. 133, pp. 483–524. Birkhäuser, Basel (1995) CrossRefGoogle Scholar
  29. 138.
    Hofer, H., Zehnder, E.: A new capacity for symplectic manifolds. In: Rabinowitz, P., Zehnder, E. (eds.) Analysis et Cetera, pp. 405–428. Academic Press, San Diego (1990) Google Scholar
  30. 139.
    Hofer, H., Zehnder, E.: Symplectic Invariants and Hamiltonian Dynamics. Birkhäuser Advanced Texts. Birkhäuser, Basel (1994) zbMATHCrossRefGoogle Scholar
  31. 153.
    José, J.V., Saletan, E.J.: Classical Dynamics. A Contemporary Approach. Cambridge University Press, Cambridge (1998) zbMATHCrossRefGoogle Scholar
  32. 165.
    Klingenberg, W.: Lectures on Closed Geodesics. Grundlehren Math. Wiss., vol. 230. Springer, Berlin (1978) zbMATHCrossRefGoogle Scholar
  33. 166.
    Kobayashi, S., Nomizu, K.: Foundations of Differential Geometry. Wiley-Interscience, New York (1963) zbMATHGoogle Scholar
  34. 168.
    Kolmogorov, A.N.: On conservation of conditionally periodic motions under small perturbations of the Hamiltonian. Dokl. Akad. Nauk SSSR 98(4), 527–530 (1954), in Russian MathSciNetzbMATHGoogle Scholar
  35. 169.
    Kolmogorov, A.N.: General theory of dynamical systems and classical mechanics. In: Proc. Int. Congr. Math., vol. 1, Amsterdam, 1954, pp. 315–333 (1957), in Russian. English translation in Abraham, R., Marsden, J.E.: Foundations of Mechanics, pp. 741–757. Benjamin-Cummings, Reading (1978) Google Scholar
  36. 170.
    Koon, W.S., Marsden, J.E.: The Hamiltonian and Lagrangian approaches to the dynamics of nonholonomic systems. Rep. Math. Phys. 40(1), 21–62 (1997) MathSciNetADSzbMATHCrossRefGoogle Scholar
  37. 181.
    Libermann, P., Marle, C.-M.: Symplectic Geometry and Analytical Mechanics. Reidel, Dordrecht (1987) zbMATHCrossRefGoogle Scholar
  38. 186.
    Liu, G., Tian, G.: Floer homology and Arnold conjecture. J. Differ. Geom. 49, 1–74 (1998) MathSciNetzbMATHGoogle Scholar
  39. 188.
    Lyapunov, A.M.: Problème général de la stabilité du mouvement. Ann. Fac. Sci. Toulouse 2, 203–474 (1907) Google Scholar
  40. 206.
    McDuff, D., Salamon, D.: Introduction to Symplectic Topology. Clarendon Press, Oxford (1998) zbMATHGoogle Scholar
  41. 209.
    Meixner, J., Schaefke, W.: Mathieusche Funktionen und Sphäroidfunktionen. Grundlehren Math. Wiss., vol. 71. Springer, Berlin (1954), in German Google Scholar
  42. 216.
    Morbidelli, A.: Modern Celestial Mechanics. Taylor & Francis, London (2002) Google Scholar
  43. 217.
    Moser, J.: On invariant curves of area preserving mappings of the annulus. Nachr. Akad. Wiss. Göttingen, Math. Phys. Kl. II, 286–294 (1962) Google Scholar
  44. 219.
    Moser, J.: A rapidly convergent iteration method and nonlinear partial differential equations I. Ann. Sc. Norm. Super. Pisa, Cl. Sci. 20(2), 265–315 (1966) zbMATHGoogle Scholar
  45. 220.
    Moser, J.: A rapidly convergent iteration method and nonlinear partial differential equations II. Ann. Sc. Norm. Super. Pisa, Cl. Sci. 20(3), 499–533 (1966) Google Scholar
  46. 224.
    Moser, J.: Periodic orbits near an equilibrium and a theorem by Alan Weinstein. Commun. Pure Appl. Math. 29, 727–747 (1976) ADSzbMATHCrossRefGoogle Scholar
  47. 225.
    Moser, J.: Proof of a generalized form of a fixed point theorem due to G.D. Birkhoff. In: Geometry and Topology. Lecture Notes in Mathematics, vol. 597, pp. 464–494. Springer, Berlin (1977) CrossRefGoogle Scholar
  48. 226.
    Moser, J.: Addendum to “Periodic orbits near an equilibrium and a theorem by Alan Weinstein”. Commun. Pure Appl. Math. 31, 529–530 (1978) CrossRefGoogle Scholar
  49. 240.
    Poincaré, H.: Méthodes Nouvelles de la Méchanique Céleste. Gauthier-Villars, Paris (1899) Google Scholar
  50. 246.
    Pöschel, J.: A lecture on the classical KAM theorem. Proc. Symp. Pure Math. 69, 707–732 (2001) Google Scholar
  51. 249.
    Rabinowitz, P.: Periodic solutions of Hamiltonian systems. Commun. Pure Appl. Math. 31, 157–184 (1978) MathSciNetCrossRefGoogle Scholar
  52. 260.
    Salamon, D.: Lectures on Floer homology. In: Eliashberg, Y., Traynor, L. (eds.) Symplectic Geometry and Topology. IAS/Park City Mathematics Series, vol. 7, pp. 143–230 (1999) Google Scholar
  53. 269.
    Schwarz, M.: Introduction to symplectic Floer homology. In: Thomas, C.B. (ed.) Contact and Symplectic Geometry. Publ. Newton Inst., vol. 8, pp. 151–170. Cambridge University Press, Cambridge (1996) Google Scholar
  54. 270.
    Sevryuk, M.B.: Some problems of KAM-theory: conditionally-periodic motion in typical systems. Russ. Math. Surv. 50(2), 341–353 (1995) MathSciNetzbMATHCrossRefGoogle Scholar
  55. 273.
    Siegel, C.L., Moser, J.K.: Lectures on Celestial Mechanics. Grundlehren Math. Wiss., vol. 187. Springer, Berlin (1971) zbMATHCrossRefGoogle Scholar
  56. 286.
    Thirring, W.: Lehrbuch der Mathematischen Physik. Band 1: Klassische Dynamische Systeme. Springer, Berlin (1977), in German Google Scholar
  57. 299.
    Viterbo, C.: A proof of the Weinstein conjecture in ℝ2n. Ann. Inst. Henri Poincaré, Anal. Non Linéaire 4, 337–357 (1987) MathSciNetzbMATHGoogle Scholar
  58. 304.
    Weinstein, A.: Normal modes for nonlinear Hamiltonian systems. Invent. Math. 20, 47–57 (1973) MathSciNetADSzbMATHCrossRefGoogle Scholar
  59. 306.
    Weinstein, A.: Periodic orbits for convex Hamiltonian systems. Ann. Math. 108, 507–518 (1978) zbMATHCrossRefGoogle Scholar
  60. 307.
    Weinstein, A.: On the hypothesis of Rabinowitz’s periodic orbit theorems. J. Differ. Equ. 33, 353–358 (1979) zbMATHCrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media Dordrecht 2013

Authors and Affiliations

  • Gerd Rudolph
    • 1
  • Matthias Schmidt
    • 1
  1. 1.Institute for Theoretical PhysicsUniversity of LeipzigLeipzigGermany

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