Hamiltonian Systems
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 Hamil
- PDF / 1,648,629 Bytes
- 64 Pages / 439.37 x 666.142 pts Page_size
- 96 Downloads / 263 Views
Hamiltonian Systems
In this chapter we start discussing the theory of Hamiltonian systems. We begin with an introduction to the subject, including the Legendre transformation and a brief discussion of linear nonholonomic systems. Next, we present three classes of examples, which will play a role in the subsequent chapters: the geodesic flow, Hamiltonian systems on Lie group manifolds and Hamiltonian systems on coadjoint orbits. We close the elementary part of this chapter by presenting the time-dependent picture. In Sect. 9.4 we investigate the structure of regular energy surfaces and discuss the problem of the existence of closed integral curves for autonomous systems. This leads us to the famous Weinstein conjecture and to some aspects of symplectic topology. We will see that a special type of symplectic invariants, called symplectic capacities, constitutes the most important technical tool for studying the Weinstein conjecture. In Sect. 9.5 we start to investigate the behaviour of a Hamiltonian system near a critical integral curve. We will see that, generically, there is a continuum of periodic integral curves nearby, constituting so called orbit cylinders. These cylinders can undergo bifurcations. We study one of these bifurcations, provided by the Lyapunov Centre Theorem, in detail. Next, we derive the so-called Birkhoff normal form both for symplectomorphisms in the neighbourhood of elliptic fixed points and for the Hamiltonian of a system near an equilibrium. This normal form implies a foliation of the phase space into invariant tori and, in the normal form approximation, the theory becomes integrable. According to the celebrated KAM theory many of the invariant tori persist the perturbation caused by taking into account the full symplectomorphism or the full Hamiltonian, respectively. Moreover, we use the Birkhoff Normal Form Theorem to prove the Birkhoff-Lewis Theorem, which yields the existence of infinitely many periodic points near a closed integral curve of a certain type on a given energy surface. Finally, we study 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. In Sect. 9.8 we deal with the stability problem of time-periodic systems with emphasis on parametric resonance and in the final section we comment on the famous Arnold G. Rudolph, M. Schmidt, Differential Geometry and Mathematical Physics, Theoretical and Mathematical Physics, DOI 10.1007/978-94-007-5345-7_9, © Springer Science+Business Media Dordrecht 2013
427
428
9
Hamiltonian Systems
conjecture about the existence of fixed points of Hamiltonian symplectomorphisms and its relation to the existence problem for closed integral curves.
9.1 Introduction A Hamiltonian system is a triple (M, ω, H ), where (M, ω) is a symplectic manifold, in the present context called the phase space, and H is a smooth function on M, called the Hamiltonian or the Hamiltonian function. In this section we restrict ourselves to autonomous1 systems.
Data Loading...
