Hamiltonian Systems

  • Gerd Rudolph
  • Matthias Schmidt
Chapter
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.
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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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