Remarks on the Lagrangian representation of bi-Hamiltonian equations

TL;DR

This paper explores the conditions under which bi-Hamiltonian PDEs admit a Lagrangian representation, focusing on systems with a hydrodynamic-type Hamiltonian operator and providing explicit examples including a 3-component WDVV system.

Contribution

It establishes a criterion linking Lagrangian representation to a generalized vector field and applies this to find explicit Lagrangian forms for third-order homogeneous Hamiltonian operators.

Findings

Lagrangian representation exists if a specific generalized vector field condition is met.

Explicit Lagrangian for a 3-component WDVV hydrodynamic system is derived.

Method applicable to various homogeneous Hamiltonian operators.

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.