Journal of Nonlinear Mathematical Physics

Volume 24, Issue Supplement 1, December 2017, Pages 121 - 145

Variational symmetries and pluri-Lagrangian systems in classical mechanics

Authors
Matteo Petrera
Institut für Mathematik, MA 7-1, Technische Universität Berlin, Str. des 17. Juni 136, 10623 Berlin, Germany. petrera@math.tu-berlin.de
Yuri B. Suris
Institut für Mathematik, MA 7-1, Technische Universität Berlin, Str. des 17. Juni 136, 10623 Berlin, Germany, suris@math.tu-berlin.de
Received 4 October 2017, Accepted 16 November 2017, Available Online 6 January 2021.
DOI
https://doi.org/10.1080/14029251.2017.1418058How to use a DOI?
Keywords
Lagrangian system, variational symmetry, Noether theorem, pluri-Lagrangian structure, integrable system
Abstract

We analyze the relation of the notion of a pluri-Lagrangian system, which recently emerged in the theory of integrable systems, to the classical notion of variational symmetry, due to E. Noether. We treat classical mechanical systems and show that, for any Lagrangian system with m commuting variational symmetries, one can construct a pluri-Lagrangian 1-form in the (m + 1)-dimensional time, whose multi-time Euler-Lagrange equations coincide with the original system supplied with m commuting evolutionary flows corresponding to the variational symmetries. We also give a Hamiltonian counterpart of this construction, leading, for any system of commuting Hamiltonian flows, to a pluri-Lagrangian 1-form with coefficients depending on functions in the phase space.

Copyright
© 2017 The Authors. Published by Atlantis Press and Taylor & Francis
Open Access
This is an open access article distributed under the CC BY-NC 4.0 license (http://creativecommons.org/licenses/by-nc/4.0/).

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Journal
Journal of Nonlinear Mathematical Physics
Volume-Issue
24 - Supplement 1
Pages
121 - 145
Publication Date
2021/01
ISSN (Online)
1776-0852
ISSN (Print)
1402-9251
DOI
https://doi.org/10.1080/14029251.2017.1418058How to use a DOI?
Copyright
© 2017 The Authors. Published by Atlantis Press and Taylor & Francis
Open Access
This is an open access article distributed under the CC BY-NC 4.0 license (http://creativecommons.org/licenses/by-nc/4.0/).

Cite this article

TY  - JOUR
AU  - Matteo Petrera
AU  - Yuri B. Suris
PY  - 2021
DA  - 2021/01
TI  - Variational symmetries and pluri-Lagrangian systems in classical mechanics
JO  - Journal of Nonlinear Mathematical Physics
SP  - 121
EP  - 145
VL  - 24
IS  - Supplement 1
SN  - 1776-0852
UR  - https://doi.org/10.1080/14029251.2017.1418058
DO  - https://doi.org/10.1080/14029251.2017.1418058
ID  - Petrera2021
ER  -