A simple way of making a Hamiltonian system into a bi-Hamiltonian one
نویسنده
چکیده
Given a Poisson structure (or, equivalently, a Hamiltonian operator) P , we show that its Lie derivative Lτ (P ) along a vector field τ defines another Poisson structure, which is automatically compatible with P , if and only if [L τ (P ), P ] = 0, where [·, ·] is the Schouten bracket. This result yields a new local description for the set of all Poisson structures compatible with a given Poisson structure P such that dim kerP ≤ 1 and leads to a remarkably simple construction of bi-Hamiltonian dynamical systems. A new description for pairs of compatible local Hamiltonian operators of Dubrovin–Novikov type is also presented.
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