A Variational Principle for Discrete Integrable Systems
For integrable systems in the sense of multidimensional consistency (MDC), we can consider the Lagrangian as a form that 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...
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| Published in: | Symmetry, Integrability and Geometry: Methods and Applications |
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| Date: | 2018 |
| Main Authors: | , |
| Format: | Article |
| Language: | English |
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Інститут математики НАН України
2018
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| Online Access: | https://nasplib.isofts.kiev.ua/handle/123456789/209531 |
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| Journal Title: | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| Cite this: | A Variational Principle for Discrete Integrable Systems / S.B. Lobb, F.W. Nijhoff // Symmetry, Integrability and Geometry: Methods and Applications. — 2018. — Т. 14. — Бібліогр.: 27 назв. — англ. |
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Digital Library of Periodicals of National Academy of Sciences of Ukraine| _version_ | 1862699720375271424 |
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| author | Lobb, S.B. Nijhoff, F.W. |
| author_facet | Lobb, S.B. Nijhoff, F.W. |
| citation_txt | A Variational Principle for Discrete Integrable Systems / S.B. Lobb, F.W. Nijhoff // Symmetry, Integrability and Geometry: Methods and Applications. — 2018. — Т. 14. — Бібліогр.: 27 назв. — англ. |
| collection | DSpace DC |
| container_title | Symmetry, Integrability and Geometry: Methods and Applications |
| description | For integrable systems in the sense of multidimensional consistency (MDC), we can consider the Lagrangian as a form that 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 also as the defining equations for the Lagrangians themselves.
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| first_indexed | 2025-12-07T16:36:33Z |
| format | Article |
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| id | nasplib_isofts_kiev_ua-123456789-209531 |
| institution | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| issn | 1815-0659 |
| language | English |
| last_indexed | 2025-12-07T16:36:33Z |
| publishDate | 2018 |
| publisher | Інститут математики НАН України |
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| spelling | Lobb, S.B. Nijhoff, F.W. 2025-11-24T10:43:21Z 2018 A Variational Principle for Discrete Integrable Systems / S.B. Lobb, F.W. Nijhoff // Symmetry, Integrability and Geometry: Methods and Applications. — 2018. — Т. 14. — Бібліогр.: 27 назв. — англ. 1815-0659 2010 Mathematics Subject Classification: 35Q51; 37K60; 39A14; 49N99 arXiv: 1312.1440 https://nasplib.isofts.kiev.ua/handle/123456789/209531 https://doi.org/10.3842/SIGMA.2018.041 For integrable systems in the sense of multidimensional consistency (MDC), we can consider the Lagrangian as a form that 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 also as the defining equations for the Lagrangians themselves. The authors would like to thank James Atkinson for helpful comments and suggestions. At the time of writing, SL was supported by the Australian Laureate Fellowship Grant #FL120100094 from the Australian Research Council. FN was partially supported by the grant EP/I038683/1 of the Engineering and Physical Sciences Research Council (EPSRC). FN is also grateful to the hospitality of the Sophus Lie Center in Nordfjordeid (Norway) during the conference on “Nonlinear Mathematical Physics: Twenty Years of JNMP” (June 4–14, 2013) where a preliminary account (joint with SL) of the results of this paper was first presented [20] before the appearance of other papers reporting similar results. en Інститут математики НАН України Symmetry, Integrability and Geometry: Methods and Applications A Variational Principle for Discrete Integrable Systems Article published earlier |
| spellingShingle | A Variational Principle for Discrete Integrable Systems Lobb, S.B. Nijhoff, F.W. |
| title | A Variational Principle for Discrete Integrable Systems |
| title_full | A Variational Principle for Discrete Integrable Systems |
| title_fullStr | A Variational Principle for Discrete Integrable Systems |
| title_full_unstemmed | A Variational Principle for Discrete Integrable Systems |
| title_short | A Variational Principle for Discrete Integrable Systems |
| title_sort | variational principle for discrete integrable systems |
| url | https://nasplib.isofts.kiev.ua/handle/123456789/209531 |
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