Lobb, SB and Nijhoff, FW (2018) A varA variational principle for discrete integrable systemsiational principle for discrete integrable systems. SIGMA, 14.
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.
Metadata
Item Type: | Article |
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Authors/Creators: |
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Copyright, Publisher and Additional Information: | © 2018, Author(s). This is an open access article under the terms of the Creative Commons Attribution License (CC BY-SA-4.0). Attribution-ShareAlike 4.0 International (CC BY-SA 4.0) |
Keywords: | variational calculus; Lagrangian multiforms; discrete integrable systems |
Dates: |
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Institution: | The University of Leeds |
Academic Units: | The University of Leeds > Faculty of Engineering & Physical Sciences (Leeds) > School of Mathematics (Leeds) > Applied Mathematics (Leeds) |
Depositing User: | Symplectic Publications |
Date Deposited: | 03 May 2018 12:10 |
Last Modified: | 25 Jun 2023 21:20 |
Status: | Published |
Publisher: | National Academy of Science of Ukraine |
Identification Number: | 10.3842/SIGMA.2018.041 |
Open Archives Initiative ID (OAI ID): | oai:eprints.whiterose.ac.uk:130379 |