Regular and Chaotic Dynamics

, Volume 19, Issue 3, pp 415–434 | Cite as

Superintegrable generalizations of the Kepler and Hook problems

  • Ivan A. Bizyaev
  • Alexey V. Borisov
  • Ivan S. Mamaev
Article

Abstract

In this paper we consider superintegrable systems which are an immediate generalization of the Kepler and Hook problems, both in two-dimensional spaces — the plane ℝ2 and the sphere S2 — and in three-dimensional spaces ℝ3 and S3. Using the central projection and the reduction procedure proposed in [21], we show an interrelation between the superintegrable systems found previously and show new ones. In all cases the superintegrals are presented in explicit form.

Keywords

superintegrable systems Kepler and Hook problems isomorphism central projection reduction highest degree polynomial superintegrals 

MSC2010 numbers

70H06 70G10 37J35 

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Copyright information

© Pleiades Publishing, Ltd. 2014

Authors and Affiliations

  • Ivan A. Bizyaev
    • 1
  • Alexey V. Borisov
    • 1
    • 2
    • 3
  • Ivan S. Mamaev
    • 1
    • 4
  1. 1.Udmurt State UniversityIzhevskRussia
  2. 2.A.A. Blagonravov Mechanical Engineering Research Institute of RASMoscowRussia
  3. 3.National Research Nuclear University “MEPhI”MoscowRussia
  4. 4.Institute of Mathematics and Mechanics of the Ural Branch of RASEkaterinburgRussia

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