Generalized Kepler problems and Euclidean Jordan algebras. (English) Zbl 1353.70042

Meng, Guowu

Mladenov, Ivaïlo M. (ed.) et al., Proceedings of the 17th international conference on geometry, integrability and quantization, Sts. Constantine and Elena (near Varna), Bulgaria, June 5–10, 2015. Sofia: Avangard Prima. 72-94 (2016).

Summary: This article is a written version of author’s lecture on generalized Kepler problems at the Conference mentioned above. It begins with a review of the Kepler problem for planetary motion and its magnetized cousins, from which a surprising relationship with Lorentz transformation emerges. Next, we give a review for Euclidean Jordan algebra and the associated universal Kepler problem. Finally, we demonstrate that, via the universal Kepler problem, a suitable Poisson realization of the conformal algebra for a simple euclidean Jordan algebra gives rise to a super integrable model that resembles the Kepler problem. In particular, we demonstrate how the Kepler problem and its magnetized cousins are obtained this way.
For the entire collection see [Zbl 1330.53003].

MSC:

70G65	Symmetries, Lie group and Lie algebra methods for problems in mechanics
17C50	Jordan structures associated with other structures
17B63	Poisson algebras
37J05	Relations of dynamical systems with symplectic geometry and topology (MSC2010)

Keywords:

Jordan algebra; Kepler problem; Lorentz transformation; Poisson bracket; principal bundle; super integrable models; symmetric cone