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

Yuri B. Suris

doi:10.3934/jgm.2013.5.365

Article Contents

2013, Volume 5, Issue 3: 365-379. Doi: 10.3934/jgm.2013.5.365

This issue Previous Article The Kadomtsev-Petviashvili hierarchy and the Mulase factorization of formal Lie groups Next Article

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

Yuri B. Suris^1,

1.
Institut für Mathematik, MA 7-2, Technische Universität Berlin, Str. des 17. Juni 136, 10623 Berlin

Received: November 30, 2012

Revised: July 31, 2013

Published: August 2013

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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 (Bäcklund 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 with the only input being the maps themselves.

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Mathematics Subject Classification: Primary: 37J05, 37J10, 37J15, 37J35, 49S05, 70H03, 70H05; Secondary: 70H06.

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