Overview

Hamiltonian systems model a number of important problems in theoretical physics, celestial mechanics, fluid dynamics etc. The stability problems, being significant for Hamiltonian systems, gave rise to sophisticated mathematical methods, see for example the books by [Guckenheimer and Holmes (1983); Arnold et al. (1996); Arnold and Givental (2001)]. Properties of Hamiltonian systems like conservation of energy (represented by a Hamiltonian function) and conservation of volume in the phase space lead to specific dynamical features.

In this chapter we study multi-parameter linear Hamiltonian systems with finite degrees of freedom. These systems are represented by Hamiltonian matrices, whose eigenvalues are situated on the complex plane symmetrically with respect to the real and imaginary axes. Due to this property, the stability of a linear Hamiltonian system can be only marginal, and the asymptotic stability can not take place. That is why the stability analysis of an equilibrium for a linearized Hamiltonian system does not guarantee the stability of a non-linear system. Nevertheless it provides the necessary stability condition. Qualitative analysis of multi-parameter linear Hamiltonian systems was done in [Galin (1975); Patera et al. (1982); Ko ak (1984)]. Bifurcation theory of eigenvalues and methods of multi-parameter stability analysis in case of linear Hamiltonian systems was given in [Mailybaev and Seyranian (1998c); Mailybaev and Seyranian (1999a); Seyranian and Mailybaev (1999)], and our presentation is mostly based on these papers.

We start the chapter with the brief introduction to dynamics and stability of Hamiltonian systems and their relationship to gyroscopic systems. Then bifurcation...