Integrable Hamiltonian Hierarchies and Lagrangian 1-Forms

Abstract

We present further developments on the Lagrangian 1-form description for one-dimensional integrable systems in both discrete and continuous levels. A key feature of integrability in this context called a closure relation will be derived from the local variation of the action on the space of independent variables. The generalised Euler-Lagrange equations and constraint equations are derived directly from the variation of the action on the space of dependent variables. This set of Lagrangian equations gives rise to a crucial property of integrable systems known as the multidimensional consistency. Alternatively, the closure relation can be obtained from generalised Stokes' theorem exhibiting a path independent property of the systems on the space of independent variables. The homotopy structure of paths suggests that the space of independent variables is simply connected. Furthermore, the Nöether charges, invariants in the context of Liouville integrability, can be obtained directly from the non-local variation of the action on the space of dependent variables.

arXiv comment42 pages, 11 figures

References in corpus (8)

• Finite extinction time for the solutions to the Ricci flow on certain three-manifolds
• Variational formulation of commuting Hamiltonian flows: multi-time Lagrangian 1-forms
• Variational symmetries and pluri-Lagrangian systems in classical mechanics
• Multi-time Lagrangian 1-forms for families of Bäcklund transformations. Toda-type systems
• Multi-time Lagrangian 1-forms for families of Bäcklund transformations. Relativistic Toda-type systems
• Discrete-time Ruijsenaars-Schneider system and Lagrangian 1-form structure
• On the Lagrangian structure of 3D consistent systems of asymmetric quad-equations
• On multidimensional consistent systems of asymmetric quad-equations