Nijenhuis Coordinates Research Articles

Lifting geometric objects to the dual of the first jet bundle of a bundle fibred over [formula omitted

We study natural lifting operations from a bundle E over R to the dual bundle of its first-jet bundle. The main purpose is to define a complete lift of a type (1,1) tensor field on E and to understand all features of its construction. Various other lifting operations of tensorial objects on E are needed for that purpose. We prove that the complete lift of a type (1,1) tensor with vanishing Nijenhuis torsion gives rise to a Poisson-Nijenhuis structure on the dual of the first-jet bundle, and discuss in detail how the construction of associated Darboux-Nijenhuis coordinates can be carried out.

Multi-Hamiltonian structures for r-matrix systems

For the rational, elliptic, and trigonometric r-matrices, we exhibit the links between three “levels” of Poisson spaces: (a) some finite-dimensional spaces of matrix-valued meromorphic functions on the complex line, (b) spaces of spectral curves and sheaves supported on them, and (c) symmetric products of a surface. We have, at each level, a linear space of compatible Poisson structures, and the maps relating the levels are Poisson. This leads in a natural way to the Nijenhuis coordinates for these spaces. At level (b), there are Hamiltonian systems on these spaces which are integrable for each Poisson structure in the family and which are such that the Lagrangian leaves are the intersections of the symplectic leaves over the Poisson structures in the family. Specific examples include many of the well-known integrable systems.

Towards a classification of natural bi-hamiltonian systems

For construction and classification of the natural integrable systems we propose to use a criterion of separability in Darboux--Nijenhuis coordinates, which can be tested without an a priori explicit knowledge of these coordinates.

Integrable natural systems in pseudoeuclidean plane

Abstract The description of all natural Hamiltonian systems in pseudoeuclidean plane is given which have an additional integral, quadratic in momenta. For polynomial potentials the recurrent formula generating bi-Hamiltonian chains is obtained. The Darboux—Nijenhuis coordinates are introduced due to which dynamic equations are integrated in quadratures. The examples of bi-Hamiltonian chains are considered.

Quasi-Bi-Hamiltonian Systems Obtained from Constrained Flows

An infinite number of families of quasi-bi-Hamiltonian (QBH) systems can be constructed from the constrained flows of soliton equations. The Nijenhuis coordinates for the QBH systems are proved to be exactly the same as the separation variables introduced by the Lax matrices for the constrained flows.

Bihamiltonian Geometry and Separation of Variables for Toda Lattices

We discuss the bihamiltonian geometry of the Toda lattice (periodic and open). Using some recent results on the separation of variables for bihamiltonian manifold, we show that these systems can be explicitly integrated via the classical Hamilton–Jacobi method in the so-called Darboux–Nijenhuis coordinates.

Quasi-Bi-Hamiltonian Systems Obtained from Constrained Flows

An infinite number of families of quasi-bi-Hamiltonian (QBH) systems can be costructed from the constrained flows of soliton equations. The Nijenhuis coordinates for the QBH systems are proved to be exactly the same as the separation variables introduced by the Lax matrices for the constrained flows.

Non-Pfaffian quasi-bi-Hamiltonian systems with two degrees of freedom

In the case of two degrees of freedom, a (non-Pfaffian) quasi-bi-Hamiltonian system with a separable integrating factor is presented (in terms of Nijenhuis coordinates) and its separability is proved. Indications are given in the case of a general integrating factor.

Bi-Hamiltonian separable chains on Riemannian manifolds

Abstract Bi-Hamiltonian separable chains in so-called Nijenhuis coordinates, which are quadratic in the momentum variables, are related with a large classes of Stackel systems.

Quasi-bi-Hamiltonian systems and separability

Two quasi--biHamiltonian systems with three and four degrees of freedom are presented. These systems are shown to be separable in terms of Nijenhuis coordinates. Moreover the most general Pfaffian quasi-biHamiltonian system with an arbitrary number of degrees of freedom is constructed (in terms of Nijenhuis coordinates) and its separability is proved.

Two degrees of freedom quasi-bi-Hamiltonian systems

Starting from the classical example of the Henon - Heiles integrable Hamiltonian system, we show that it admits a slightly different formulation from the classical bi-Hamiltonian system. We introduce the general notion of a quasi-bi-Hamiltonian system (QBHS) and study some of its basic properties only in the case of two degrees of freedom; in particular, we give the general form of this structure in the Nijenhuis coordinates which are constructed using the eigenvalues of the Nijenhuis operator of the system and show how they serve to separate the Hamilton - Jacobi equation. In the last section, we look at the problem of the semi-local existence of such structures.