Involution Analysis of the Partial Differential Equations Characterising Hamiltonian Vector Fields

Reference: Journal of Mathematical Physics, 44 (2003) 1173-1182

Description: This paper discusses some results of Bender, Dunne and Mead concerning a certain underdetermined linear system of partial differential equations associated with a Poisson manifold. I extend their results from Lie-Poisson structures to arbitrary Poisson structures. Bender et al. derived their system by analysing special power series solutions of an evolution equation. I show that the system simply represents the conditions on the components of a vector field for the field to be Hamiltonian with respect to the given Poisson structure. By performing an involution analysis of the system, I can correct an error of Bender et al. in the case of degenerate Poisson structures.

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