Canonical Forms for Bihamiltonian Systems

Abstract

BiHamiltonian systems were first defined in the fundamental paper of Magri, [5], which deduced the integrability of many soliton equations from the fact that they could be written in Hamiltonian form in two distinct ways. More recently, the classical completely integrable Hamiltonian systems of ordinary differential equations, such as the Toda lattice and rigid body, have been shown to be biHamiltonian systems. However, recent results of Brouzet, [1], extended by Fernandes, [3], indicate that there are global, topological obstructions to the existence of a biHamiltonian structure for a general completely integrable Hamiltonian system. The connection between biHamiltonian structures and R-matrices, [10], which provide solutions to the classical Yang-Baxter equation, has given additional impetus to their study.