Stefan Rauch-Wojciechowski

Constrained flows of integrable PDEs and bi-Hamiltonian structure of the Garnier system

Abstract A new reduction procedure for integrable multi-Hamiltonian PDEs is introduced. It leads to a multi-Hamiltonian description of the resulting finite dimensional dynamical systems. The bi-Hamiltonian structure of the Garnier system is studied in some detail.

How to construct finite‐dimensional bi‐Hamiltonian systems from soliton equations: Jacobi integrable potentials

A systematic method of constructing finite‐dimensional integrable systems starting from a bi‐Hamiltonian hierarchy of soliton equations is introduced. The existence of two Hamiltonian structures of the hierarchy leads to a bi‐Hamiltonian formulation of the resulting finite‐dimensional systems. The case of coupled KdV hierarchies is studied in detail. A surprising connection with separable Jacobi potentials is uncovered and described.

Bi-Hamiltonian formulation of the Hénon-Heiles system and its multidimensional extensions

Abstract We prove that a (slightly) generalized Henon-Heiles system is equivalent to a fifth order Hamiltonian evolution equation with a third order Hamiltonian operator. This equivalence makes it possible to use the machinery of restricted flows of soliton hierarchies in order to (a) find natural extensions of integrable cases of the Henon-Heiles system, and (b) determine (in the KdV case) a bi-Hamiltonian formulation for the (extended) Henon-Heiles system and prove its complete integrability.

Quasi-Lagrangian systems of Newton equations

Systems of Newton equations of the form q=−12A−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 partial differential equation 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 foun...

A generalized Henon–Heiles system and related integrable Newton equations

A detailed description is given of integrable cases of the generalized Henon–Heiles systems which differs from the standard H–H ones by the term α/q22. Their connection with fifth‐order one‐component soliton equations is discussed. Lax representations are constructed, and the bi‐Hamiltonian formulation of dynamics is given. It is also shown that the gH–H system can be mapped onto another system of Newton equations with a nonstandard Hamiltonian structure.

A bi-Hamiltonian formulation for separable potentials and its application to the Kepler problem and the Euler problem of two centers of gravitation

Abstract We give here a bi-Hamiltonian formulation for natural Hamiltonian systems separable in generalized elliptic, generalized parabolic and spherical coordinates. It requires the use of an extra variable so that the phase space is odd dimensional and the Poisson brackets are degenerate. The general results are illustrated by several examples: in particular the Kepler problem and the Euler problem of two centers of gravitation are discussed.

Two newton decompositions of staionary flows of KdV and Harry Dym hierarchies

We show that each stationary flow of the KdV and Harry Dym hierarchies of soliton equations, which are (2m+1)-st order ODEs (m=0,1…), has two parametrisations as a set of Newton equations with velocity-independent forces. Forces are potential and these Newton equations follow from a Lagrangian function with an inefinite kinetic energy term. These two parametrisations are canonically inequivalent and give rise to new bihamiltonian structures in classical mechanics. Lax representations for these Newton equations are found.

What an Effective Criterion of Separability says about the Calogero Type Systems

Abstract In [15] we have proved a 1-1 correspondence between all separable coordinates on R n (according to Kalnins and Miller [9]) and systems of linear PDEs for separable potentials V (q). These PDEs, after introducing parameters reflecting the freedom of choice of Euclidean reference frame, serve as an effective criterion of separability. This means that any V (q) satisfying these PDEs is separable and that the separation coordinates can be determined explicitly. We apply this criterion to Calogero systems of particles interacting with each other along a line.

Newton representation for stationary flows of the KdV hierarchy

Abstract We introduce a new parametrization for stationary flows of the KdV hierarchy which turns them into a set of Newton equations. In the case of a fifth order KdV flow the equivalence with the integrable case of the Henon-Heiles system leads to a bi-Hamiltonian formulation of the corresponding Newton equations. A simple generalization of its second Poisson bracket yields a new family of integrable potentials in two dimensions.

Two families of nonstandard Poisson structures for Newton equations

Two 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.