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Regul. Chaotic Dyn., 2014, Volume 19, Issue 3, Pages 415–434 (Mi rcd163)  

This article is cited in 9 scientific papers (total in 9 papers)

Superintegrable Generalizations of the Kepler and Hook Problems

Ivan A. Bizyaeva, Alexey V. Borisovabc, Ivan S. Mamaevad

a Udmurt State University, Universitetskaya 1, Izhevsk, 426034 Russia
b A. A. Blagonravov Mechanical Engineering Research Institute of RAS, Bardina str. 4, Moscow, 117334, Russia
c National Research Nuclear University “MEPhI”, Kashirskoye shosse 31, Moscow, 115409, Russia
d Institute of Mathematics and Mechanics of the Ural Branch of RAS, S. Kovalevskaja str. 16, Ekaterinburg, 620990, Russia

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 $\mathbb{R}^2$ and the sphere $S^2$ — and in three-dimensional spaces $\mathbb{R}^3$ and $S^3$. 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

Funding Agency Grant Number
Russian Foundation for Basic Research 13-01-12462-ofi_m
The work of A.V. Borisov was done within the framework of the State assignment of the Udmurt State University “Regular and Chaotic Dynamics”. The work of I.S.Mamaev was supported by the grant of the RFBR 13-01-12462-ofi m, and the work of I.A.Bizyaev was supported by the grant of the RFBR 14-01-00395-a.


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Document Type: Article
MSC: 70H06, 70G10, 37J35
Received: 27.03.2014
Language: English

Citation: Ivan A. Bizyaev, Alexey V. Borisov, Ivan S. Mamaev, “Superintegrable Generalizations of the Kepler and Hook Problems”, Regul. Chaotic Dyn., 19:3 (2014), 415–434

Citation in format AMSBIB
\by Ivan~A.~Bizyaev, Alexey~V.~Borisov, Ivan~S.~Mamaev
\paper Superintegrable Generalizations of the Kepler and Hook Problems
\jour Regul. Chaotic Dyn.
\yr 2014
\vol 19
\issue 3
\pages 415--434

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    This publication is cited in the following articles:
    1. I. A. Bizyaev, “Ob odnom obobschenii sistem tipa Kalodzhero”, Nelineinaya dinam., 10:2 (2014), 209–212  mathnet
    2. Andrey V. Tsiganov, “Killing Tensors with Nonvanishing Haantjes Torsion and Integrable Systems”, Regul. Chaotic Dyn., 20:4 (2015), 463–475  mathnet  crossref  mathscinet  zmath  adsnasa  elib
    3. I. A. Bizyaev, A. V. Borisov, I. S. Mamaev, “Hamiltonization of elementary nonholonomic systems”, Russ. J. Math. Phys., 22:4 (2015), 444–453  crossref  mathscinet  zmath  isi  scopus
    4. A. Ballesteros, A. Blasco, F. J. Herranz, F. Musso, “An integrable Hénon–Heiles system on the sphere and the hyperbolic plane”, Nonlinearity, 28:11 (2015), 3789–3801  crossref  mathscinet  zmath  isi  scopus
    5. A. V. Tsiganov, “Two integrable systems with integrals of motion of degree four”, Theoret. and Math. Phys., 186:3 (2016), 383–394  mathnet  crossref  crossref  mathscinet  adsnasa  isi  elib
    6. Alexey V. Borisov, Ivan S. Mamaev, Ivan A. Bizyaev, “The Spatial Problem of 2 Bodies on a Sphere. Reduction and Stochasticity”, Regul. Chaotic Dyn., 21:5 (2016), 556–580  mathnet  crossref  mathscinet  zmath  elib
    7. M. F. Ranada, “Superintegrable systems with a position dependent mass: Kepler-related and oscillator-related systems”, Phys. Lett. A, 380:27-28 (2016), 2204–2210  crossref  mathscinet  zmath  isi  scopus
    8. Galliano Valent, “Superintegrable Models on Riemannian Surfaces of Revolution with Integrals of any Integer Degree (I)”, Regul. Chaotic Dyn., 22:4 (2017), 319–352  mathnet  crossref
    9. Shengda Hu, Manuele Santoprete, “Suslov Problem with the Clebsch–Tisserand Potential”, Regul. Chaotic Dyn., 23:2 (2018), 193–211  mathnet  crossref
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