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 3dimensional 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.

Quantum variational principle and quantum multiform structure: The case of quadratic Lagrangians

“…In fact, the standard variational principle on (2) produces two copies of (1). In order to regain precisely the linearised KdV equation, we must make use of the multiform variational principle introduced by Lobb and Nijhoff [5,12]. (1) can be consistently embedded into a multidimensional lattice, with directions labelled by subscripts i, j, k. Across an elementary plaquette in the i − j plane, (1) takes the form:…”

“…In the variational principle proposed in [12], the action is defined as the sum of Lagrangians on elementary plaquettes across a 2-dimensional surface σ, embedded in the multidimensional space. To derive the equations of motion, we then demand the action be stationary not only under the variation of the field variables u, but also under the variation of the surface σ itself.…”

“…where we have used the shorthand L i j (u) := L(u, u i , u j ; p i , p j ), and the final equality in (5) holds only when we apply (4). According to [12], such a system must be described by a Lagrangian of the form…”

“…A recent development in understanding discrete integrable systems with commuting flows has been the Lagrangian multiform theory [12,5,9,10,29,11,14]. A system with two or more commuting, discrete flows can be described by a Lagrangian 1-form structure, which provides a way to obtain a simultaneous system of equations for a single dependent variable from a variational principle.…”

“…Further extensions and deepening understanding of these results were obtained in a number of papers, cf. [12][13][14].…”