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arXiv · 1610.01817

Remarks on the Lagrangian representation of bi-Hamiltonian equations

M. V. Pavlov·R. F. Vitolo
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Abstract

The Lagrangian representation of multi-Hamiltonian PDEs has been introduced by Y. Nutku and one of us (MVP). In this paper we focus on systems which are (at least) bi-Hamiltonian by a pair $A_1$, $A_2$, where $A_1$ is a hydrodynamic-type Hamiltonian operator. We prove that finding the Lagrangian representation is equivalent to finding a generalized vector field $\tau$ such that $A_2=L_\tau A_1$. We use this result in order to find the Lagrangian representation when $A_2$ is a homogeneous third-order Hamiltonian operator, although the method that we use can be applied to any other homogeneous Hamiltonian operator. As an example we provide the Lagrangian representation of a WDVV hydrodynamic-type system in $3$ components.

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M. V. Pavlov, R. F. Vitolo. 2016-10-06. Remarks on the Lagrangian representation of bi-Hamiltonian equations. https://doi.org/10.1016/j.geomphys.2016.10.013

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