An r-matrix formalism is applied to the construction of the integrable lattice systems and their bi-Hamiltonian structure. Miura-like gauge transformations between the hierarchies are also investigated. In the end the ladder of linear maps between generated hierarchies is established and described.
Systems of Newton equations of the formq = − 1 2 A −1 (q)∇k with an integral of motion quadratic in velocities are studied. These equations generalize the potential case (when A=I, the identity matrix) and they admit a curious quasi-Lagrangian formulation which differs from the standard Lagrange equations by the plus sign between terms. A theory of such quasi-Lagrangian Newton (qLN) systems having two functionally independent integrals of motion is developed with focus on two-dimensional systems. Such systems admit a bi-Hamiltonian formulation and are proved to be completely integrable by embedding into five-dimensional integrable systems. They are characterized by a linear, second-order PDE which we call the fundamental equation. Fundamental equations are classified through linear pencils of matrices associated with qLN systems. The theory is illustrated by two classes of systems: separable potential systems and driven systems. New separation variables for driven systems are found. These variables are based on sets of non-confocal conics. An effective criterion for existence of a qLN formulation of a given system is formulated and applied to dynamical systems of the Hénon-Heiles type.
Note on the Poisson structure of the damped oscillatorTwo families of nonstandard two-dimensional Poisson structures for systems of Newton equations are studied. They are closely related either with separable systems or with the so-called quasi-Lagrangian systems. A theorem characterizing the general form of bi-Hamiltonian formulation for separable systems in two and in n dimensions is formulated and proved.
We propose a general scheme of constructing of soliton hierarchies from finite dimensional Stäckel systems and related separation relations. In particular, we concentrate on the simplest class of separation relations, called Benenti class, i.e. certain Stäckel systems with quadratic in momenta integrals of motion.
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