Hamiltonian Systems of Few Degrees of Freedom

  • Tassos Bountis
  • Haris Skokos
Chapter
First Online:
Part of the Springer Series in Synergetics book series (SSSYN, volume 10)

Abstract

In Chap. 2 we provide first an elementary introduction to some simple examples of Hamiltonian systems of one and two degrees of freedom. We describe the essential features of phase space plots and focus on the concepts of periodic and quasiperiodic motions. We then address the questions of integrability and solvability of the equations, first for linear and then for nonlinear problems. We present the important integrability criteria of Painlevé analysis in complex time and show, on the non-integrable Hénon-Heiles model, how chaotic orbits arise on a Poincaré Surface of Section of the dynamics in phase space. Using the example of a periodically driven Duffing oscillator, we explain that chaos is connected with the intersection of invariant manifolds and describe how these intersections can be analytically studied by the perturbation approach of Mel’nikov theory.

Keywords

Hamiltonian System Invariant Manifold Unstable Manifold Homoclinic Orbit Heteroclinic Orbit 
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.
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Copyright information

© Springer-Verlag Berlin Heidelberg 2012

Authors and Affiliations

  • Tassos Bountis
    • 1
  • Haris Skokos
    • 2
    • 3
  1. 1.Department of MathematicsUniversity of PatrasPatrasGreece
  2. 2.MPI for the Physics of Complex SystemsDresdenGermany
  3. 3.Physics DepartmentAristotle University of ThessalonikiThessalonikiGreece

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