The Lie Algebra Structure of Degenerate Hamiltonian and Bi-Hamiltonian Systems

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Abstract
A generalization of Noether's theorem is obtained via an extension of the well-known Poisson bracket formalism. It is shown that degenerate closed forms yield Lie algebra homomorphisms between vector fields and covector fields. A similar result holds for operators working in the opposite way. Application of these Lie algebra homomorphisms to a dynamical system having two (degenerate) Hamiltonian formulations yields a selfmap in the space of infinitesimal generators of one-parameter symmetry groups of this system. These Hamiltonian formulations are not assumed to constitute a Hamiltonian pair (in the sense of Gelfand-Dorfman). Thus infinite-dimensional symmetry groups for a wider class of equations can be constructed. Several new equations are shown to admit infinite dimensional symmetry groups.