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arXiv · 1207.3042

Poisson bracket on 1-forms and evolutionary partial differential equations

Alessandro Arsie·Paolo Lorenzoni
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Abstract

We introduce a bracket on 1-forms defined on ${\cal J}^{\infty}(S^1, \mathbb{R}^n)$, the infinite jet extension of the space of loops and prove that it satisfies the standard properties of a Poisson bracket. Using this bracket, we show that certain hierarchies appearing in the framework of $F$-manifolds with compatible flat connection $(M, \nabla, \circ)$ are Hamiltonian in a generalized sense. Moreover, we show that if a metric $g$ compatible with $\nabla$ is also invariant with respect to $\circ$, then this generalized Hamiltonian set-up reduces to the standard one.

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Alessandro Arsie, Paolo Lorenzoni. 2012-07-12. Poisson bracket on 1-forms and evolutionary partial differential equations. https://doi.org/10.1088/1751-8113%2F45%2F47%2F475208

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