Systematic construction of non-autonomous Hamiltonian equations of Painlevé-type. I. Frobenius integrability

Abstract

This article is the first one in a suite of three articles exploring connections between dynamical systems of Stäckel-type and of Painlevé- type. In this article we present a deformation of autonomous Stäckel-type systems to non-autonomous Frobenius integrable systems. First, we consider quasi-Stäckel systems with quadratic in momenta Hamiltonians containing separable potentials with time dependent coefficients and then we present a procedure of deforming these equations to non-autonomous Frobenius integrable systems. Then, we present a procedure of deforming quasi-Stäckel systems with so called magnetic separable potentials to non-autonomous Frobenius integrable systems. We also provide a complete list of all $2$- and $3$-dimensional Frobenius integrable systems, both with ordinary and with magnetic potentials, that originate in our construction. Further, we prove the equivalence between both classes of systems. Finally we show how Painlevé equations $P_I-P_{IV}$ can be derived from our scheme.

32 pages

References in corpus (4)

• Natural coordinates for a class of Benenti systems
• Hamiltonian Structure of PI Hierarchy
• Non-Homogeneous Hydrodynamic Systems and Quasi-Stäckel Hamiltonians
• Non-autonomous Henon-Heiles system from Painleve class

Cited by in corpus (1)

• Knizhnik-Zamolodchikov equations and integrable hyperbolic Landau-Zener models