Duality between integrable Stackel systems

Abstract

For the Stackel 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 Stackel's systems with two degrees of freedom the 2x2 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.

LaTeX2e, 18 pages

References in corpus (1)

• Dynamical boundary conditions for integrable lattices

Cited by in corpus (17)

• On the Drach superintegrable systems
• Integrable Euler top and nonholonomic Chaplygin ball
• The Maupertuis principle and canonical transformations of the extended phase space
• On maximally superintegrable systems
• Canonical transformations of the extended phase space, Toda lattices and Stackel family of integrable systems
• Leonard Euler: addition theorems and superintegrable systems
• Normal forms for pseudo-Riemannian 2-dimensional metrics whose geodesic flows admit integrals quadratic in momenta
• An integrable hierarchy with a perturbed Henon-Heiles system
• Addition theorems and the Drach superintegrable systems
• One invariant measure and different Poisson brackets for two nonholonomic systems
• The Lax pairs for the Holt system
• Transformation of the Stackel matrices preserving superintegrability
• Lax Representations for Separable Systems from Benenti Class
• Canonical transformations of the time for the Toda lattice and the Holt system
• On the superintegrable Richelot systems
• On one integrable system with a cubic first integral
• Stäckel Equivalence of Non-Degenerate Superintegrable Systems, and Invariant Quadrics