Variational formulation of commuting Hamiltonian flows: multi-time Lagrangian 1-forms

TL;DR

This paper develops a variational multi-time Lagrangian framework for integrable systems, linking multi-time Euler-Lagrange equations to commuting Hamiltonian flows and symplectic maps, with implications for Lax representations.

Contribution

It introduces a comprehensive Lagrangian theory for multi-time integrable systems, establishing the equivalence between multi-time Euler-Lagrange equations and commuting Hamiltonian flows, and connects spectrality with Lagrangian form closure.

Findings

Multi-time Euler-Lagrange equations are equivalent to commuting Hamiltonian flows.

Closure of the Lagrangian 1-form corresponds to involutivity of Hamilton functions.

Spectrality property of symplectic maps relates to the Lagrangian 1-form being closed.

Abstract

Recently, Lobb and Nijhoff initiated the study of variational (Lagrangian) structure of discrete integrable systems from the perspective of multi-dimensional consistency. In the present work, we follow this line of research and develop a Lagrangian theory of integrable one-dimensional systems. We give a complete solution of the following problem: one looks for a function of several variables (interpreted as multi-time) which delivers critical points to the action functionals obtained by integrating a Lagrangian 1-form along any smooth curve in the multi-time. The Lagrangian 1-form is supposed to depend on the first jet of the sought-after function. We derive the corresponding multi-time Euler-Lagrange equations and show that, under the multi-time Legendre transform, they are equivalent to a system of commuting Hamiltonian flows. Involutivity of the Hamilton functions turns out to be equivalent to closeness of the Lagrangian 1-form on solutions of the multi-time Euler-Lagrange equations. In the discrete time context, the analogous extremal property turns out to be characteristic for systems of commuting symplectic maps. For one-parameter families of commuting symplectic maps (B\"acklund transformations), we show that their spectrality property, introduced by Kuznetsov and Sklyanin, is equivalent to the property of the Lagrangian 1-form to be closed on solutions of the multi-time Euler-Lagrange equations, and propose a procedure of constructing Lax representations starting from the maps themselves.