Variational symmetries and pluri-Lagrangian systems in classical mechanics

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.

25 pp

References in corpus (5)

• Faddeev-Volkov solution of the Yang-Baxter Equation and Discrete Conformal Symmetry
• On the Lagrangian structure of integrable hierarchies
• Multi-time Lagrangian 1-forms for families of Bäcklund transformations. Relativistic Toda-type systems
• Discrete pluriharmonic functions as solutions of linear pluri-Lagrangian systems
• Billiards in confocal quadrics as a pluri-Lagrangian system

Cited by in corpus (14)

• Variational symmetries and Lagrangian multiforms
• Lagrangian multiforms for Kadomtsev-Petviashvili (KP) and the Gelfand-Dickey hierarchy
• Variational symmetries and pluri-Lagrangian structures for integrable hierarchies of PDEs
• Continuum limits of pluri-Lagrangian systems
• Multiform description of the AKNS hierarchy and classical r-matrix
• Hamiltonian multiform description of an integrable hierarchy
• Lagrangian multiforms on coadjoint orbits for finite-dimensional integrable systems
• Lagrangian multiforms on Lie groups and non-commuting flows
• Semi-discrete Lagrangian 2-forms and the Toda hierarchy
• Hamiltonian structures for integrable hierarchies of Lagrangian PDEs
• Integrable Hamiltonian Hierarchies and Lagrangian 1-Forms
• Quantum integrability: Lagrangian 1-form case
• On the geometry of Lagrangian one-forms
• Lagrangian multiforms and dispersionless integrable systems