# American Institute of Mathematical Sciences

September  2013, 5(3): 365-379. doi: 10.3934/jgm.2013.5.365

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

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

Received  December 2012 Revised  August 2013 Published  September 2013

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.
Citation: Yuri B. Suris. Variational formulation of commuting Hamiltonian flows: Multi-time Lagrangian 1-forms. Journal of Geometric Mechanics, 2013, 5 (3) : 365-379. doi: 10.3934/jgm.2013.5.365
##### 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. Google Scholar [2] A. I. Bobenko and Yu. B. Suris, Integrable systems on quad-graphs,, International Mathematics Research Notices, 2002 (2002), 573. doi: 10.1155/S1073792802110075. Google Scholar [3] A. I. Bobenko and Yu. B. Suris, "Discrete Differential Geometry. Integrable Structure,", Graduate Studies in Mathematics, (2008). Google Scholar [4] A. I. Bobenko and Yu. B. Suris, On the Lagrangian structure of integrable quad-equations,, Letters in Mathematical Physics, 92 (2010), 17. doi: 10.1007/s11005-010-0381-9. Google Scholar [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). doi: 10.1088/1751-8113/46/27/275204. Google Scholar [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. doi: 10.1088/0305-4470/31/9/012. Google Scholar [7] S. B. Lobb and F. W. Nijhoff, Lagrangian multiforms and multidimensional consistency,, Journal of Physics A: Mathematical and Theoretical, 42 (2009). doi: 10.1088/1751-8113/42/45/454013. Google Scholar [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). doi: 10.1088/1751-8113/43/7/072003. Google Scholar [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). doi: 10.1088/1751-8113/42/47/472002. Google Scholar [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. doi: 10.1007/BF02352494. Google Scholar [11] F. W. Nijhoff, Lax pair for the Adler (lattice Krichever-Novikov) system,, Physics Letters A, 297 (2002), 49. doi: 10.1016/S0375-9601(02)00287-6. Google Scholar [12] J. Roels and A. Weinstein, Functions whose Poisson brackets are constants,, Journal of Mathematical Physics, 12 (1971), 1482. doi: 10.1063/1.1665760. Google Scholar [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. Google Scholar [14] Yu. B. Suris, "The Problem of Integrable Discretization: Hamiltonian Approach,", Progress in Mathematics, (2003). doi: 10.1007/978-3-0348-8016-9. Google Scholar [15] M. Wadati and M. Toda, Bäcklund transformation for the exponential lattice,, Journal of the Physical Society of Japan, 39 (1975), 1196. doi: 10.1143/JPSJ.39.1196. Google Scholar [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). doi: 10.1088/1751-8113/44/36/365203. Google Scholar

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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. Google Scholar [2] A. I. Bobenko and Yu. B. Suris, Integrable systems on quad-graphs,, International Mathematics Research Notices, 2002 (2002), 573. doi: 10.1155/S1073792802110075. Google Scholar [3] A. I. Bobenko and Yu. B. Suris, "Discrete Differential Geometry. Integrable Structure,", Graduate Studies in Mathematics, (2008). Google Scholar [4] A. I. Bobenko and Yu. B. Suris, On the Lagrangian structure of integrable quad-equations,, Letters in Mathematical Physics, 92 (2010), 17. doi: 10.1007/s11005-010-0381-9. Google Scholar [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). doi: 10.1088/1751-8113/46/27/275204. Google Scholar [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. doi: 10.1088/0305-4470/31/9/012. Google Scholar [7] S. B. Lobb and F. W. Nijhoff, Lagrangian multiforms and multidimensional consistency,, Journal of Physics A: Mathematical and Theoretical, 42 (2009). doi: 10.1088/1751-8113/42/45/454013. Google Scholar [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). doi: 10.1088/1751-8113/43/7/072003. Google Scholar [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). doi: 10.1088/1751-8113/42/47/472002. Google Scholar [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. doi: 10.1007/BF02352494. Google Scholar [11] F. W. Nijhoff, Lax pair for the Adler (lattice Krichever-Novikov) system,, Physics Letters A, 297 (2002), 49. doi: 10.1016/S0375-9601(02)00287-6. Google Scholar [12] J. Roels and A. Weinstein, Functions whose Poisson brackets are constants,, Journal of Mathematical Physics, 12 (1971), 1482. doi: 10.1063/1.1665760. Google Scholar [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. Google Scholar [14] Yu. B. Suris, "The Problem of Integrable Discretization: Hamiltonian Approach,", Progress in Mathematics, (2003). doi: 10.1007/978-3-0348-8016-9. Google Scholar [15] M. Wadati and M. Toda, Bäcklund transformation for the exponential lattice,, Journal of the Physical Society of Japan, 39 (1975), 1196. doi: 10.1143/JPSJ.39.1196. Google Scholar [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). doi: 10.1088/1751-8113/44/36/365203. Google Scholar
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