Deforming Lie algebras to Frobenius integrable nonautonomous Hamiltonian systems. (English) Zbl 1489.37079
Summary: Motivated by the theory of Painlevé equations and associated hierarchies, we study nonautonomous Hamiltonian systems that are Frobenius integrable. We establish sufficient conditions under which a given finite-dimensional Lie algebra of Hamiltonian vector fields can be deformed into a time-dependent Lie algebra of Frobenius integrable vector fields spanning the same distribution as the original algebra. The results are applied to quasi-Stäckel systems from [K. Marciniak and M. Błaszak, SIGMA, Symmetry Integrability Geom. Methods Appl. 13, Paper 077, 15 p. (2017; Zbl 1387.70019)].
MSC:
| 37J65 | Nonautonomous Hamiltonian dynamical systems (Painlevé equations, etc.) |
| 37J37 | Relations of finite-dimensional Hamiltonian and Lagrangian systems with Lie algebras and other algebraic structures |
| 37J35 | Completely integrable finite-dimensional Hamiltonian systems, integration methods, integrability tests |
| 17B80 | Applications of Lie algebras and superalgebras to integrable systems |
Keywords:
Liouville integrability; Lie algebras; Frobenius integrability; separable systems; quasi-Stäckel systemsCitations:
Zbl 1387.70019References:
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