Duality of Hamiltonian and Lagrangian formulations for integrable systems

TL;DR

This paper explores the duality between Hamiltonian and Lagrangian formulations in integrable systems, introducing Hamiltonian potential variables and constructing new Lagrangian multiforms for key models.

Contribution

It generalizes the Hamiltonian-Lagrangian duality using potential variables and provides new Lagrangian multiforms for several integrable systems.

Findings

Introduced Hamiltonian potential variables to map Hamiltonian to symplectic operators.

Constructed new Lagrangian multiforms for KdV and dispersionless limits.

Presented the first Lagrangian multiforms for polytropic gas dynamics and the constant astigmatism equation.

Abstract

We introduce the concept of Hamiltonian potential variables to map Hamiltonian operators into symplectic operators in a dual space. This generalises the classical trick of switching to a potential variable to obtain a Lagrangian density for the Korteweg-de Vries (KdV) equation. Building on this concept, we present the Lagrangian structure for bi-Hamiltonian systems, discuss the Lenard scheme in the symplectic formalisms, and apply this to construct pairs of Lagrangian multiforms. We discuss the key model of the KdV equation and some dispersionless limits of it. We present a pair of Lagrangian multiforms for these equations, one of which is new. We also consider the examples of polytropic gas dynamics and the constant astigmatism equation, for which no Lagrangian multiforms were previously known.