Chapter 16 Extended hamiltonian systems

Abstract

This chapter discusses Hamiltonian partial differential wave equations that are defined on unbounded spatial domains, a class of so-called “extended Hamiltonian systems.” The contrast in dynamics between Hamiltonian systems of extended type and those of compact type is striking. Compact Hamiltonian systems arising, for example, from finite dimensional Hamiltonian systems or Hamiltonian partial differential equations (PDE), governing an evolutionary process, defined on a bounded spatial domain, are systems governed by finite or infinite systems of ordinary differential equations (ODE) with a discrete set of frequencies. In contrast, extended Hamiltonian systems, arising from Hamiltonian PDEs, are systems involving continuous as well as discrete spectra of frequencies. Stable states are expected to be asymptotically stable; states initially nearby the unperturbed state remain close and even converge to it in an appropriate metric. Since the flow is in an infinite-dimensional space, this does not contradict the Hamiltonian character of the phase flow that in finite dimensional spaces preserves volume. Convergence to an asymptotic state occurs through a mechanism of radiating energy to infinity. It is also possible that some states of the system are long-lived metastable states.