Variational formulation of commuting Hamiltonian flows: Multi-time Lagrangian 1-forms

Yuri B. Suris

Research output: Contribution to journalArticlepeer-review

26 Scopus citations


Recently, Lobb and Nijhoff initiated the study of variational (Lagrangian) structure of discrete integrable systems from the perspective of multidimensional consistency. In the present work, we follow this line of research and develop a Lagrangian theory of integrable one-dimensional systems. We give a complete solution of the following problem: one looks for a function of several variables (interpreted as multi-time) which delivers critical points to the action functionals obtained by integrating a Lagrangian 1-form along any smooth curve in the multi-time. The Lagrangian 1-form is supposed to depend on the first jet of the sought-after function. We derive the corresponding multitime Euler-Lagrange equations and show that, under the multi-time Legendre transform, they are equivalent to a system of commuting Hamiltonian flows. Involutivity of the Hamilton functions turns out to be equivalent to closeness of the Lagrangian 1-form on solutions of the multi-time Euler-Lagrange equations. In the discrete time context, the analogous extremal property turns out to be characteristic for systems of commuting symplectic maps. For one-parameter families of commuting symplectic maps (Bäcklund transformations), we show that their spectrality property, introduced by Kuznetsov and Sklyanin, is equivalent to the property of the Lagrangian 1-form to be closed on solutions of the multi-time Euler-Lagrange equations, and propose a procedure of constructing Lax representations with the only input being the maps themselves.

Original languageEnglish
Pages (from-to)365-379
Number of pages15
JournalJournal of Geometric Mechanics
Issue number3
StatePublished - Sep 2013
Externally publishedYes


  • Baecklund transformations
  • Commuting Hamiltonian flows
  • Commuting symplectic maps
  • Discrete Lagrangian mechanics
  • Euler-Lagrange equations
  • Hamiltonian mechanics
  • Integrable systems
  • Lagrangian 1-form
  • Lagrangian mechanics
  • Multitime
  • Spectrality


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