2012/12/31 by Yuri B. Suris · 2 citations
Physics and Astronomy · Mathematics · #math-ph #math.MP #math.SG #nlin.SI #msc:37J05 #msc:37J10 #msc:37J35 #msc:49S05 #msc:70H03 #msc:70H25 #msc:70H06
paper · pdf · doi:10.3934/jgm.2013.5.365
published as Journal of Geometric Mechanics, 2013, Volume 5, Issue 3, pp.: 365 -379 · 15 pp., 2 figs. In v2, we added the relation of the presented theory to the Kuznetsov-Sklyanin's spectrality
arxiv created 2013/01/24 · arxiv updated 2014/03/13
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 starting from the maps themselves.