Duality between integrable Stäckel systems

Abstract

For the Stäckel family of the integrable systems a non-canonical transformation of the time variable is considered. This transformation may be associated to the ambiguity of the Abel map on the corresponding hyperelliptic curve. For some Stäckel's systems with two degrees of freedom the 2 × 2 Lax representations and the dynamical r-matrix algebras are constructed. As an examples the Henon-Heiles systems, integrable Holt potentials and the integrable deformations of the Kepler problem are discussed in detail.

The Maupertuis Principle and Canonical Transformations of the Extended Phase Space

“…We have some examples of integrable systems related with one Lagrangian submanifold [23,38,39], which satisfy to the usual Kowalevski-Painlevé property and the "weak" Painlevé property, respectively. These systems are related with the different parametrisations of one geometric object.…”

Untitled

“…In our case the Stäckel matrix S (2.12) is the lowest block of the transpose Brill-Noether matrix U C (2.13)) and, therefore, there are canonical coordinates in which equations of motion (2.8) are the Newton equations [16,17].…”

“…According to [10,16,17] the generic 2 × 2 Lax matrices for the uniform Stäckel system are constructed using Hamiltonian H 1 and the generating function e(λ) of the separated variables only…”

“…Here e t = {H 1 , e} and function w(λ) is given by w(λ) = φ(λ)e(λ) −1 M N , where φ(λ) is a parametric function and [ξ] M N is the linear combinations of the following Laurent projections [16,17] […”

“…According to [17] the Stäckel systems associated with the different blocks of a common Brill-Noether matrix are related by canonical transformation of the time.…”