### Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

SIGMA 14 (2018), 041, 18 pages      arXiv:1312.1440      https://doi.org/10.3842/SIGMA.2018.041
Contribution to the Special Issue on Recent Advances in Quantum Integrable Systems

### A Variational Principle for Discrete Integrable Systems

Sarah B. Lobb a and Frank W. Nijhoff b
a) NSW Department of Education, Sydney NSW 2000, Australia
b) School of Mathematics, University of Leeds, Leeds LS2 9JT, UK

Received April 01, 2017, in final form April 26, 2018; Published online May 03, 2018

Abstract
For integrable systems in the sense of multidimensional consistency (MDC) we can consider the Lagrangian as a form, which is closed on solutions of the equations of motion. For 2-dimensional systems, described by partial difference equations with two independent variables, MDC allows us to define an action on arbitrary 2-dimensional surfaces embedded in a higher dimensional space of independent variables, where the action is not only a functional of the field variables but also the choice of surface. It is then natural to propose that the system should be derived from a variational principle which includes not only variations with respect to the dependent variables, but also with respect to variations of the surface in the space of independent variables. Here we derive the resulting system of generalized Euler-Lagrange equations arising from that principle. We treat the case where the equations are 2 dimensional (but which due to MDC can be consistently embedded in higher-dimensional space), and show that they can be integrated to yield relations of quadrilateral type. We also derive the extended set of Euler-Lagrange equations for 3-dimensional systems, i.e., those for equations with 3 independent variables. The emerging point of view from this study is that the variational principle can be considered as the set of equations not only encoding the equations of motion but as the defining equations for the Lagrangians themselves.

Key words: variational calculus; Lagrangian multiforms; discrete integrable systems.

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References

1. Adler V.E., Bobenko A.I., Suris Yu.B., Classification of integrable equations on quad-graphs. The consistency approach, Comm. Math. Phys. 233 (2003), 513-543, nlin.SI/0202024.
2. Bobenko A.I., Suris Yu.B., On the Lagrangian structure of integrable quad-equations, Lett. Math. Phys. 92 (2010), 17-31, arXiv:0912.2464.
3. Bobenko A.I., Suris Yu.B., Discrete pluriharmonic functions as solutions of linear pluri-Lagrangian systems, Comm. Math. Phys. 336 (2015), 199-215, arXiv:1403.2876.
4. Boll R., Petrera M., Suris Yu.B., Multi-time Lagrangian 1-forms for families of Bäcklund transformations: Toda-type systems, J. Phys. A: Math. Theor. 46 (2013), 275204, 26 pages, arXiv:1302.7144.
5. Boll R., Petrera M., Suris Yu.B., What is integrability of discrete variational systems?, Proc. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci. 470 (2014), 20130550, 15 pages, arXiv:1307.0523.
6. Boll R., Petrera M., Suris Yu.B., Multi-time Lagrangian 1-forms for families of Bäcklund transformations. Relativistic Toda-type systems, J. Phys. A: Math. Theor. 48 (2015), 085203, 28 pages, arXiv:1408.2405.
7. Boll R., Petrera M., Suris Yu.B., On the variational interpretation of the discrete KP equation, in Advances in Discrete Differential Geometry, Editor A.I. Bobenko, Springer, Berlin, 2016, 379-405, arXiv:1506.00729.
8. Boll R., Suris Yu.B., On the Lagrangian structure of 3D consistent systems of asymmetric quad-equations, J. Phys. A: Math. Theor. 45 (2012), 115201, 18 pages, arXiv:1108.0016.
9. Cadzow J.A., Discrete calculus of variations, Internat. J. Control 11 (1970), 393-407.
10. Capel H.W., Nijhoff F.W., Papageorgiou V.G., Complete integrability of Lagrangian mappings and lattices of KdV type, Phys. Lett. A 155 (1991), 377-387.
11. Hietarinta J., Joshi N., Nijhoff F.W., Discrete systems and integrability, Cambridge Texts in Applied Mathematics, Cambridge University Press, Cambridge, 2016.
12. King S.D., Nijhoff F.W., Quantum variational principle and quantum multiform structure: the case of quadratic Lagrangians, arXiv:1702.08709.
13. Lobb S.B., Nijhoff F.W., Lagrangian multiforms and multidimensional consistency, J. Phys. A: Math. Theor. 42 (2009), 454013, 18 pages, arXiv:0903.4086.
14. Lobb S.B., Nijhoff F.W., Lagrangian multiform structure for the lattice Gel'fand-Dikii hierarchy, J. Phys. A: Math. Theor. 43 (2010), 072003, 11 pages, arXiv:0911.1234.
15. Lobb S.B., Nijhoff F.W., Quispel G.R.W., Lagrangian multiform structure for the lattice KP system, J. Phys. A: Math. Theor. 42 (2009), 472002, 11 pages, arXiv:0906.5282.
16. Logan J.D., First integrals in the discrete variational calculus, Aequationes Math. 9 (1973), 210-220.
17. Maeda S., Canonical structure and symmetries for discrete systems, Math. Japon. 25 (1980), 405-420.
18. Maeda S., Extension of discrete Noether theorem, Math. Japon. 26 (1981), 85-90.
19. Maeda S., Lagrangian formulation of discrete systems and concept of difference space, Math. Japon. 27 (1982), 345-356.
20. Nijhoff F.W., New variational principle for integrable systems, Talk at ''Nonlinear Mathematical Physics: Twenty Years of JNMP'' (Norway, June 4-14, 2013), available at http://staff.www.ltu.se/~norbert/JNMP-Conference-2013/JNMP-conference-2013.html.
21. Sahadevan R., Rasin O.G., Hydon P.E., Integrability conditions for nonautonomous quad-graph equations, J. Math. Anal. Appl. 331 (2007), 712-726, nlin.SI/0611019.
22. Suris Yu.B., Variational formulation of commuting Hamiltonian flows: multi-time Lagrangian 1-forms, J. Geom. Mech. 5 (2013), 365-379, arXiv:1212.3314.
23. Suris Yu.B., Vermeeren M., On the Lagrangian structure of integrable hierarchies, in Advances in Discrete Differential Geometry, Editor A.I. Bobenko, Springer, Berlin, 2016, 347-378, arXiv:1510.03724.
24. Xenitidis P., Nijhoff F.W., Lobb S.B., On the Lagrangian formulation of multidimensionally consistent systems, Proc. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci. 467 (2011), 3295-3317, arXiv:1008.1952.
25. Yoo-Kong S., Calogero-Moser type systems, associated KP systems, and Lagrangian structures, Ph.D. Thesis, University of Leeds, 2011.
26. Yoo-Kong S., Lobb S.B., Nijhoff F.W., Discrete-time Calogero-Moser system and Lagrangian 1-form structure, J. Phys. A: Math. Theor. 44 (2011), 365203, 39 pages, arXiv:1102.0663.
27. Yoo-Kong S., Nijhoff F.W., Discrete-time Ruijsenaars-Schneider system and Lagrangian 1-form structure, arXiv:1112.4576.