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

SIGMA 14 (2018), 041, 18 pages      arXiv:1312.1440
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

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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