Variational formulation of commuting Hamiltonian flows: multi-time Lagrangian 1-forms. (English) Zbl 1330.37057
Summary: Recently, S. Lobb and F. Nijhoff [J. Phys. A, Math. Theor. 42, No. 45, Article ID 454013, 18 p. (2009; Zbl 1196.37117)] initiated the study of variational (Lagrangian) structure of discrete integrable systems from the perspective of multi-dimensional 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 multi-time 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 V. B. Kuznetsov and E. K. Sklyanin [J. Phys. A, Math. Gen. 31, No. 9, 2241–2251 (1998; Zbl 0951.37041)], 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.
MSC:
37J35 | Completely integrable finite-dimensional Hamiltonian systems, integration methods, integrability tests |
37J05 | Relations of dynamical systems with symplectic geometry and topology (MSC2010) |
37J10 | Symplectic mappings, fixed points (dynamical systems) (MSC2010) |
37J15 | Symmetries, invariants, invariant manifolds, momentum maps, reduction (MSC2010) |
70H03 | Lagrange’s equations |
70H05 | Hamilton’s equations |
70H06 | Completely integrable systems and methods of integration for problems in Hamiltonian and Lagrangian mechanics |