Classification, Casimir Invariants, and Stability of Lie-Poisson Systems

Abstract

We classify Lie-Poisson brackets that are formed from Lie algebra extensions. The problem is relevant because many physical systems owe their Hamiltonian structure to such brackets. A classification involves reducing all brackets to a set of normal forms, independent under coordinate transformations, and is achieved with the techniques of Lie algebra cohomology. For extensions of order less than five, we find that the number of normal forms is small and they involve no free parameters. A special extension, known as the Leibniz extension, is shown to be the unique ``maximal'' extension. We derive a general method of finding Casimir invariants of Lie-Poisson bracket extensions. The Casimir invariants of all brackets of order less than five are explicitly computed, using the concept of `coextension.' We obtain the Casimir invariants of Leibniz extensions of arbitrary order. We also offer some physical insight into the nature of the Casimir invariants of compressible reduced magnetohydrodynamics. We make use of the methods developed to study the stability of extensions for given classes of Hamiltonians. This helps to elucidate the distinction between semidirect extensions and those involving cocycles. For compressible reduced magnetohydrodynamics, we find the cocycle has a destabilizing effect on the steady-state solutions.