Variational symmetries and pluri-Lagrangian systems in classical mechanics
- 10.1080/14029251.2017.1418058How to use a DOI?
- Lagrangian system; variational symmetry; Noether theorem; pluri-Lagrangian structure; integrable system
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.
- © 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 - 2021/01/06 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 - 10.1080/14029251.2017.1418058 ID - Petrera2021 ER -