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Periodic Orbits of Hamiltonian Systems

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Encyclopedia of Complexity and Systems Science

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Definition

The study of periodicmotions is very important in the investigations of natural phenomena. Inparticular the Hamiltonian formulation of the laws of motion hasbeen able to formalize and solve many fundamental problems inmechanics and dynamical systems. This paper is focused ona selection of results in the study of periodic motions inHamiltonian systems. We shall consider local problems(e.?g., stability and continuation/bifurcation) and alsothe application of variational methods to study the existence inthe large. Throughout the paper we give some details of themethods and proofs.

Introduction

Periodic motions and behaviors in Nature have always beenof interest to mankind. All phenomena that have some cyclicnature have captured our attention because they are a signand a clue for regularity. Therefore, they are indicationsof the possibility of understanding the laws of Nature. Sincethe second century BC, Greek philosophers and astronomers havelooked into the possibility of...

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Abbreviations

Hamiltonian:

are called all those dynamical systems whose equations of motion form a vector field X H defined on a symplectic manifold (\( { \mathcal{P},\omega } \)), and X H is given by \( { i_{X_H}\omega=\text{d} H } \), where \( { H\colon\mathcal{P}\rightarrow \mathbb{R} } \) is the Hamiltonian function.

Poisson systems:

These are dynamical systems whose vector field X H can be described through a Poisson structure (Poisson brackets ) defined on the ring of differentiable functions on a given manifold that is not necessarily symplectic (see “Hamiltonian Equations”). Note that on any symplectic manifold there is a natural Poisson structure such that any Hamiltonian system admits a Poisson formulation, but the contrary is false. The Poisson formulation of the dynamics is a generalization of the Hamiltonian one.

A periodic orbit:

\( { \phi(.) } \) is a solution of the equations of motion that repeats itself after a certain time \( { T > 0 } \) called a period, that is, \( { \phi(t+T)=\phi(t) } \) for every t.

Poincaré section/map:

Given a periodic orbit \( { \phi(.) } \) a Poincaré section is a hyperplane S intersecting the curve \( { \{\phi(t)\colon t\in [0,T)\} } \) transversely. The associated Poincaré map ? maps neighborhoods of S into itself by following the orbit \( { \phi(.) } \) (see Definition 10).

A Hamiltonian system with symmetry:

is a Hamiltonian system in which there is a group G acting on \( { \mathcal{P} } \), i.?e., there is a map \( { \Phi\colon G\times\mathcal{P}\mapsto \mathcal{P} } \), with F preserving the Hamiltonian and the symplectic form.

Relative periodic orbit:

Let G be a symmetry group for the dynamics. A path \( { \phi(.) } \) is a relative periodic orbit if solves the equations of motion and repeats itself up to a group action after a certain time \( { T > 0 } \), that is, \( { \phi(t+T)=\Phi_g(\phi(t)) } \) for every t and for some \( { g\in G } \).

Continuation:

Continuation is a procedure based on the implicit function theorem (IFT) that allows one to extend the solution of an equation for different values of the parameters. Let \( { f(x,\epsilon)=0 } \) be an equation in \( { x\in\mathbb{R}^n } \) where f is differentiable and \( { \epsilon\geq 0 } \) a parameter. Assume that \( { f(x_0,0)=0 } \); a curve \( { x(\epsilon) } \) is called a continued solution if \( { x(0)=x_0 } \) and \( { f(x(\epsilon),\epsilon)=0 } \) for some \( { \epsilon\geq 0 } \). In general \( { x(\epsilon) } \) exists whenever the IFT can be applied, that is, if \( { D_x f(x,\epsilon) } \) is invertible at (\( { x_0,0 } \)).

Liapunov–Schmidt reduction:

Let f be a function on a Banach space. Liapunov–Schmidt reduction is a procedure that allows one to study \( { f(x,\epsilon)=0 } \) under the condition that the kernel of Df is not empty but it is finite-dimensional.

Variational principles:

The principles which aim to translate the problem of solving the equations of motion of a dynamical system (e.?g., Hamiltonian systems) into the problem of finding the critical points of certain functionals defined on spaces of all possible trajectories of the given system.

Bibliography

  1. Abraham R, Marsden J (1978) Foundations of Mechanics. Benjamin-Cummings,Reading

    MATH  Google Scholar 

  2. Ambrosetti A, Coti V (1994) Zelati Periodic Solutions of singular LagrangianSystems. Birkhäuser, Boston

    Google Scholar 

  3. AmbrosettiA, Prodi G (1993) A primer on Nonlinear Analysis. CUP, Cambridge UK

    Google Scholar 

  4. Arms JM, Cushman R, Gotay MJ (1991) A universal reduction procedure forHamiltonian group actions. In: Ratiu TS (ed) The Geometry of Hamiltonian Systems. Springer, New York,pp 33–51

    Google Scholar 

  5. ArnoldVI (1989) Mathematical Methods of Classical Mechanics. Springer, New York

    Google Scholar 

  6. Benci V (1984) Normal modes of a Lagrangian system constrained ina potential well. Annales de l'IHP section C tome 5(1):379–400

    Google Scholar 

  7. Benci V (1984) Closed geodesics for the Jacobi metric and periodic solutions ofprescribed energy of natural Hamiltonian systems. Annales de l'IHP section C tome 5(1):401–412

    Google Scholar 

  8. Benci V (1986) Periodic solutions for Lagrangian systems on a compactmanifold. J Diff Eq 63:135–161

    MathSciNet  MATH  Google Scholar 

  9. Birkhoff GD (1927) Dynamical Systems with two degrees of freedom. Trans Am MathSoc 18:199–300

    MathSciNet  Google Scholar 

  10. BirkhoffGD (1927) Dynamical Systems. Colloq. Publ. 9. Amer. Math. Soc.,Providence RI

    Google Scholar 

  11. Bolotin SV, Kozlov VV (1978) Libration in systems with many degrees offreedom. J Appl Math Mech 42:256–261

    MathSciNet  Google Scholar 

  12. BottR, Tu W (1995) Differential forms in algebraic topology. Springer, NewYork

    Google Scholar 

  13. Bridges T (2006) Canonical multi-symplectic structure on the totalexterior algebra bundle. Proc R Soc A 462:1531–1551

    MathSciNet  ADS  MATH  Google Scholar 

  14. BrownKS (1982) Cohomology of groups. Springer, New York

    MATH  Google Scholar 

  15. Chang K (1991) Infinite dimensional Morse theory and multiple solutionproblems. Birkhäuser, Boston

    Google Scholar 

  16. CushmanR, Bates L (1997) Global aspects of classicalintegrable systems. Birkhäuser, Boston

    MATH  Google Scholar 

  17. Cushmann R, Sadowski D, Efstathiou K (2005) No polar coordinates. In: MontaldiJ, Ratiu T (eds) Geometric Mechanics and Symmetry: the PeyresqLectures. Cambridge University Press, Cambridge UK

    Google Scholar 

  18. Dell Antonio G (1995) Agomenti scelti di meccanica SISSA/ISASTrieste. ref. ILAS/FM-19/1995

    Google Scholar 

  19. Doedel EJ, Keller HB, Kernvez JP (1991) Numerical analysis and control ofbifurcation problems: (I) Bifurcation in finite dimensions. Int J Bifurcat Chaos 3(1):493–520

    Google Scholar 

  20. Doedel EJ, Keller HB, Kernvez JP (1991) Numerical analysis and control ofbifurcation problems: (II) Bifurcation in infinite dimensions. Int J Bifurcat Chaos 4(1):745–772

    Google Scholar 

  21. Ekeland I (1990) Convexity methods in Hamiltonian mechanics. Springer, Berlin

    MATH  Google Scholar 

  22. Farber M (2004) Topology of Closed One-Forms. AMS, Providence

    MATH  Google Scholar 

  23. Gallavotti G (2001) Quasi periodic motions from Hipparchus to Kolmogorov. RendMat Acc Lincei s. 9, 12:125–152

    Google Scholar 

  24. Gluck H, Ziller W (1983) Existence of periodic motions of conservativesystems. In: Bombieri E (ed) Seminar on MinimalSubmanifolds. Princeton University Press, Princeton NJ

    Google Scholar 

  25. GolubitskyM, Marsden JE, Stewart I, Dellnitz M (1995) The constrainedLiapunov–Schmidt procedure and periodic orbits. FieldsInstitute Communications 4. Fields Institute, Toronto,pp 81–127

    Google Scholar 

  26. Gordon W (1975) Conservative dynamical systems involving strong force. Trans AmMath Soc 204:113–135

    MATH  Google Scholar 

  27. Guillemin V, Sternberg S (1984) Symplectic techniques in Physics. CambridgeUniversity Press, Cambridge UK

    MATH  Google Scholar 

  28. Jost J (1995) Riemannian geometry and geometric analysis. Springer, Berlin

    MATH  Google Scholar 

  29. KlingenbergW (1978) Lectures on closed geodesics. Springer, Berlin

    MATH  Google Scholar 

  30. Kozlov VV (1985) Calculus of variations in the large and classicalmechanics. Russ Math Surv 40:37–71

    MATH  Google Scholar 

  31. Kuznetsov YA (2004) Elements of Applied Bifurcation Theory. Springer, Berlin

    MATH  Google Scholar 

  32. LeimkuhlerB, Reich S (2005) Simulating Hamiltonian Dynamics. CUP, Cambridge UK

    Google Scholar 

  33. Mawhin J, Willhelm M (1990) Critical point theory and Hamiltoniansystems. Springer, Berlin

    Google Scholar 

  34. Marsden J, Ratiu TS, Scheurle J (2000) Reduction theory and theLagrange–Routh equations. J Math Phys 41:3379–3429

    MathSciNet  ADS  MATH  Google Scholar 

  35. McCord C, Montaldi J, Roberts M, Sbano L (2003) Relative Periodic Orbits ofSymmetric Lagrangian Systems. Proc. Equadiff. World Scientific, Singapore

    Google Scholar 

  36. Meyer K, Hall GR (1991) Introduction to Hamiltonian Systems and the N-BodyProblem. Springer, Berlin

    Google Scholar 

  37. Meyer K (1999) Periodic solutions of the N-body problem. LNM, vol 1719. Springer, Berlin

    Google Scholar 

  38. Montaldi J, Buono PL, Laurent F (2005) Poltz Symmetric HamiltonianBifurcations. In: Montaldi J, Ratiu T (eds) Geometric Mechanicsand Symmetry: the Peyresq Lectures. Cambridge University Press,Cambridge UK

    Google Scholar 

  39. Montaldi J, Roberts M, Stewart I (1988) Periodic solutions near equilibria ofsymmetric Hamiltonian systems. Phil Trans R Soc Lon A 325:237–293

    MathSciNet  ADS  MATH  Google Scholar 

  40. Montaldi J, Roberts M, Stewart I (1990) Existence of nonlinear normal modes ofsymmetric Hamiltonian systems. Nonlinearity 3:695–730

    MathSciNet  ADS  MATH  Google Scholar 

  41. Montaldi J, Roberts M, Stewart I (1990) Stability of nonlinear normal modes ofsymmetric Hamiltonian systems. Nonlinearity 3:731–772

    MathSciNet  ADS  MATH  Google Scholar 

  42. Munoz–AlmarazFJ, Freire E, Galan J, Doedel E, Vanderbauwhede (2003) A Continuation of periodic orbits in conservative and Hamiltonian systems. Physica D 181:1–38

    Google Scholar 

  43. Novikov SP (1982) The Hamiltonian formalism and a multivalued analogue ofMorse theory. Russ Math Surv 37:1–56

    MATH  Google Scholar 

  44. Offin D (1987) A Class of Periodic Orbits in ClassicalMechanics. J Diff Equ 66:9–117

    MathSciNet  Google Scholar 

  45. Ortega J, Ratiu T (1998) Singular Reduction of Poisson Manifolds. Lett MathPhys 46:359–372

    MathSciNet  MATH  Google Scholar 

  46. Poincaré H (1956) Le Méthodes Nouvelles de la MécaniqueCéleste. Gauthiers-Villars, Paris

    Google Scholar 

  47. Rabinowitz PH (1986) Minimax Methods in Critical Point Theory with Applicationsto Differential Equations. In: CBMS Reg. Conf. Ser. No. 56, Amer. Math. Soc., Providence, RI

    Google Scholar 

  48. Ratiu T, Sousa Dias E, Sbano L, Terra G, Tudora R (2005) A crushcourse. In: Montaldi J, Ratiu T (eds) geometric mechanics inGeometric Mechanics and Symmetry: the Peyresq Lectures. CambridgeUniversity Press, Cambridge UK

    Google Scholar 

  49. SandersJ, Verhulst F, Murdock J (2007) Averaging Methods in NonlinearDynamical Systems. Applied Mathematical Sciences, vol 59. Springer, Berlin

    Google Scholar 

  50. Schmah T (2007) A cotangent bundle slice theorem. Diff Geom Appl25:101–124

    MathSciNet  MATH  Google Scholar 

  51. Seifert H (1948) Periodische Bewegungen mechanischer Systeme. Math Z51:197–216

    MathSciNet  MATH  Google Scholar 

  52. Simó C (2001) New families of solutions in N-body problems. In: European Congress of Mathematics, vol I (Barcelona, 2000), pp 101–115, Progr. Math.,201. Birkhäuser, Basel

    Google Scholar 

  53. Struwe M (1990) Variational methods. Springer, Berlin

    MATH  Google Scholar 

  54. Verhulst F (1990) Nonlinear Dynamical Equations and DynamicalSystems. Springer, Berlin

    MATH  Google Scholar 

  55. Weinstein A (1973) Normal Modes for Nonlinear Hamiltonian Systems. InventionesMath 20:47–57

    ADS  MATH  Google Scholar 

  56. Weinstein A (1978) Bifurcations and Hamilton Principle. Math Z159:235–248

    MathSciNet  MATH  Google Scholar 

  57. Whitehead GW (1995) Elements of homotopy theory, 3rd edn. Springer, Berlin

    Google Scholar 

  58. Wulf C, Roberts M (2002) Hamiltonian Systems Near Relative PeriodicOrbits. Siam J Appl Dyn Syst 1(1):1–43

    MathSciNet  Google Scholar 

  59. Yakubovich VA, Starzhinsky VM (1975) Linear Differential Equations withPeriodic coefficients, vol 1:2. Wiley, UK

    MATH  Google Scholar 

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Acknowledgments

The author would like to thank Heinz Hanßmann for his critical and thorough reading of the manuscript andFerdinand Verhulst for his useful suggestions.

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Authors and Affiliations

  1. Mathematics Institute, University of Warwick, Warwick, UK

    Luca Sbano

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  1. Luca Sbano
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Editors and Affiliations

  1. RAMTECH LIMITED, 122 Escalle Lane, Larkspur, CA, 94939, USA

    Robert A. Meyers Ph. D. (Editor-in-Chief) (Editor-in-Chief)

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Sbano, L. (2009). Periodic Orbits of Hamiltonian Systems. In: Meyers, R. (eds) Encyclopedia of Complexity and Systems Science. Springer, New York, NY. https://doi.org/10.1007/978-0-387-30440-3_391

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