The Gauss-Knörrer map for the Rosochatius dynamical system

The Jacobi-Rosochatius Problem on an Ellipsoid: the Lax Representations and Billiards

“…, n. This is a commutative set of functions, both with respect to the canonical Poisson brackets {·, ·} and the Dirac-Poisson bracket {·, ·} D . Let us note that a possible alternative approach in the construction of the Lax representations for the Jacobi-Rosochatius problem (4.10) is by using the Lax representations of the Neumann problem (e.g., see [41]) and the well known correspondence between the Neumann problem and the geodesic flow on an ellipsoid [31,28]. An algebro-geometric study of the Neumann system on a sphere S n with the addition of the Rosochatius potential is given in [18].…”

“…The Jacobi problem on the ellipsoid (1.9) is invariant with respect to the standard T n+1 -action on C n+1 . 1 It is well known (e.g., see [36,28]), that the reduced flow can be naturally considered as a system describing the motion of a material point on the ellipsoid (1.1) under the influence of the Hook and the Rosochatius potentials [40] (1. 10) V (x) = σ 2…”

Some new integrable systems constructed from the bi-Hamiltonian systems with pure differential Hamiltonian operators

“…In [15], we showed that the Kupershmidt deformation (2) of KdV equation is equivalent to the Rosochatius deformation of KdV equation with self-consistent sources, and presented the bi-Hamiltonian structure for the Kupershmidt deformation (2). The conjecture is then proved in [16] that the Kupershmidt deformation of a bi-Hamiltonian system is itself bi-Hamiltonian.…”

Consecutive Rosochatius Deformations of the Garnier System and the Hénon-Heiles System

“…In 1985, Wojciechowski gained an analogy system (called Garnier-Rosochatius system) for the Garnier system [8,9]. Later in 1999, based on the Deift technique and a well-known theorem that the Gauss map transforms the Neumann system to the Jacobi system, Kubo et al constructed the analogy system for the Jacobi system or the geodesic flow equation on the ellipsoid [10][11][12]. In 2007, one of the authors (Zhou) generalized the Rosochatius deformations of the constrained soliton flows [13], and then the method has been extended to construct the integrable deformations of the symplectic maps [14] and the soliton equations with self-consistent sources [15].…”