Multi-Lagrangians for Integrable Systems

TL;DR

This paper introduces a novel method to construct multiple local Lagrangians for integrable systems with multi-Hamiltonian structures, expanding the variational formulations and simplifying descriptions of certain physical models.

Contribution

It develops a general scheme to generate multiple Lagrangians for integrable equations using recursion operators and extends the variational approach to physical fields like density and velocity.

Findings

Constructed 2N-1 local Lagrangians for N-fold Hamiltonian structures.

Increased to 3N-2 Lagrangians with invertible Miura transformation.

Derived a local Lagrangian for 1+1D polytropic gas dynamics.

Abstract

We propose a general scheme to construct multiple Lagrangians for completely integrable non-linear evolution equations that admit multi- Hamiltonian structure. The recursion operator plays a fundamental role in this construction. We use a conserved quantity higher/lower than the Hamiltonian in the potential part of the new Lagrangian and determine the corresponding kinetic terms by generating the appropriate momentum map. This leads to some remarkable new developments. We show that nonlinear evolutionary systems that admit $N$-fold first order local Hamiltonian structure can be cast into variational form with $2N-1$ Lagrangians which will be local functionals of Clebsch potentials. This number increases to $3N-2$ when the Miura transformation is invertible. Furthermore we construct a new Lagrangian for polytropic gas dynamics in $1+1$ dimensions which is a {\it local} functional of the physical field variables, namely density and velocity, thus dispensing with the necessity of introducing Clebsch potentials entirely. This is a consequence of bi-Hamiltonian structure with a compatible pair of first and third order Hamiltonian operators derived from Sheftel's recursion operator.