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: December 2012

Revised: August 2013

Published: September 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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References

[1]	V. E. Adler, A. I. Bobenko and Yu. B. Suris, Classification of integrable equations on quad-graphs. The consistency approach, Commun. Math. Phys., 233 (2003), 513-543.
[2]	A. I. Bobenko and Yu. B. Suris, Integrable systems on quad-graphs, International Mathematics Research Notices, 2002, No. 11, 573-611.doi: 10.1155/S1073792802110075.
[3]	A. I. Bobenko and Yu. B. Suris, "Discrete Differential Geometry. Integrable Structure," Graduate Studies in Mathematics, Vol. 98. American Mathematical Society, Providence, RI, 2008. xxiv + 404 pp.
[4]	A. I. Bobenko and Yu. B. Suris, On the Lagrangian structure of integrable quad-equations, Letters in Mathematical Physics, 92 (2010), 17-31.doi: 10.1007/s11005-010-0381-9.
[5]	R. Boll, M. Petrera and Yu. B. Suris, Multi-time Lagrangian 1-forms for families of Bäcklund transformations. Toda-type systems, Journal of Physics A: Mathematical and Theoretical, 46 (2013), 275204, 26 pp.doi: 10.1088/1751-8113/46/27/275204.
[6]	V. B. Kuznetsov and E. K. Sklyanin, On Bäcklund transformations for many-body systems, Journal of Physics A: Mathematical and General, 31 (1998), 2241-2251.doi: 10.1088/0305-4470/31/9/012.
[7]	S. B. Lobb and F. W. Nijhoff, Lagrangian multiforms and multidimensional consistency, Journal of Physics A: Mathematical and Theoretical, 42 (2009), 454013.doi: 10.1088/1751-8113/42/45/454013.
[8]	S. B. Lobb and F. W. Nijhoff, Lagrangian multiform structure for the lattice Gel'fand-Dikii hierarchy, Journal of Physics A: Mathematical and Theoretical, 43 (2010), 072003.doi: 10.1088/1751-8113/43/7/072003.
[9]	S. B. Lobb, F. W. Nijhoff and G. R. W. Quispel, Lagrangian multiform structure for the lattice KP system, Journal of Physics A: Mathematical and Theoretical, 42 (2009), 472002.doi: 10.1088/1751-8113/42/47/472002.
[10]	J. Moser and A. P. Veselov, Discrete versions of some classical integrable systems and factorization of matrix polynomials, Communications in Mathematical Physics, 139 (1991), 217-243.doi: 10.1007/BF02352494.
[11]	F. W. Nijhoff, Lax pair for the Adler (lattice Krichever-Novikov) system, Physics Letters A, 297 (2002), 49-58.doi: 10.1016/S0375-9601(02)00287-6.
[12]	J. Roels and A. Weinstein, Functions whose Poisson brackets are constants, Journal of Mathematical Physics, 12 (1971), 1482-1486.doi: 10.1063/1.1665760.
[13]	N. Román-Roy, Á.M. Rey, M. Salgado and S. Vilariño, On the $k$-symplectic, $k$-cosymplectic and multisymplectic formalisms of classical field theories, J. Geom. Mech., 3 (2011), 113-137.
[14]	Yu. B. Suris, "The Problem of Integrable Discretization: Hamiltonian Approach," Progress in Mathematics, Vol. 219. Basel: Birkhäuser, 2003, xxi + 1070 pp.doi: 10.1007/978-3-0348-8016-9.
[15]	M. Wadati and M. Toda, Bäcklund transformation for the exponential lattice, Journal of the Physical Society of Japan, 39 (1975), 1196-1203.doi: 10.1143/JPSJ.39.1196.
[16]	S. Yoo-Kong, S. Lobb and F. W. Nijhoff, Discrete-time Calogero-Moser system and Lagrangian 1-form structure, Journal of Physics A: Mathematical and Theoretical, 44 (2011), 365203.doi: 10.1088/1751-8113/44/36/365203.