VARIATIONAL FORMULATION OF COMMUTING HAMILTONIAN FLOWS: MULTI-TIME LAGRANGIAN 1-FORMS

被引:22
|
作者
Suris, Yuri B. [1 ]
机构
[1] Tech Univ Berlin, Inst Math, MA 7 2, D-10623 Berlin, Germany
来源
JOURNAL OF GEOMETRIC MECHANICS | 2013年 / 5卷 / 03期
关键词
Lagrangian mechanics; discrete Lagrangian mechanics; Hamiltonian mechanics; Euler-Lagrange equations; multitime; Lagrangian; 1-form; commuting Hamiltonian flows; commuting symplectic maps; spectrality; Baecklund transformations; integrable systems; INTEGRABLE SYSTEMS; QUAD-GRAPHS; EQUATIONS; LATTICE;
D O I
10.3934/jgm.2013.5.365
中图分类号
O29 [应用数学];
学科分类号
070104 ;
摘要
Recently, Lobb and Nijhoff initiated the study of variational (Lagrangian) structure of discrete integrable systems from the perspective of multidimensional 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 (Backlund 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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页码:365 / 379
页数:15
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