Remarks on the Lagrangian representation of bi-Hamiltonian equations

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.