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The Lie algebra structure of degenerate Hamiltonian and bi-Hamiltonian systems. (English) Zbl 1098.37540
Summary: 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 self-map 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.
MSC:
37K30Relations of infinite-dimensional systems with algebraic structures