A Variational Principle for Discrete Integrable Systems

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.

References in corpus (9)

• Lagrangian multiforms and multidimensional consistency
• Variational formulation of commuting Hamiltonian flows: multi-time Lagrangian 1-forms
• Lagrangian multiform structure for the lattice Gel'fand-Dikii hierarchy
• What is integrability of discrete variational systems?
• On the Lagrangian structure of integrable hierarchies
• Multi-time Lagrangian 1-forms for families of Bäcklund transformations. Toda-type systems
• Multi-time Lagrangian 1-forms for families of Bäcklund transformations. Relativistic Toda-type systems
• Discrete-time Ruijsenaars-Schneider system and Lagrangian 1-form structure
• On the Lagrangian structure of 3D consistent systems of asymmetric quad-equations

Cited by in corpus (3)

• Variational symmetries and pluri-Lagrangian structures for integrable hierarchies of PDEs
• On the Lagrangian multiform structure of the extended lattice Boussinesq system
• Lagrangian multiform structure of discrete and semi-discrete KP systems