Some remarks on lagrangian and hamiltonian formalisms, related to infinite-dimensional dynamical systems with symmetries

Lax Integrable Supersymmetric Hierarchies on Extended Phase Spaces

“…The expression (19) generalizes the result obtained in the papers [10,11,12] for the Lie algebra of integral-differential operators. The existence of the Bäcklund transformation (19) makes it possible to formulate the following theorem.…”

The Bargmann type reduction for some Lax integrable two-dimensional generalization of the relativistic Toda lattice

“…with taking into account the results obtained in the papers [18], [19] we can state the existence of the smooth by Frechet functionsĥ (τ) ,ĥ (T) ∈ D(M 4 ), which satisfy the relations (26) and (27) correspondingly. Then for the functions h (τ) :=ĥ (τ)…”

“…The relationships (23) describe all periodic and quasiperiodic solutions of the system (8), (9) Proof. The exact symplectic structure on the invariant subspace M 4 N ⊂ M 4 can be found by means of the discrete analog [18], [19], [20] of the Gelfand-Dikii relationship on the functional manifold M 4 in the same manner as has been done in the paper [19] for the subspaces of critical points of local conservation laws.…”

“…, ς N }. Thus, for every N ∈ N the invariant subspace M 4 N H c ⊂ M 4 is diffeomorphic to the subspaceM 4 N ⊂M 4 when ς i = λ i , i = 1, N. By means of the discrete analog of the Gelfand-Dikii differential relationship [18], [19], [20] forL N ∈ D(M 4 ) such as…”

“…c Hentosh O.Ye., 2015 symplectic structure on the invariant subspace can be found by means of the Gelfand-Dikii type relationship [5], [19] for the differential of the related Lagrangian function on a suitably extended phase space. The discrete analog of the Gelfand-Dikii relationship has been considered in [18], [19], [20].…”