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

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_m14-01-00395-a 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.

DOI: https://doi.org/10.1134/S1560354714030095

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Document Type: Article
MSC: 70H06, 70G10, 37J35
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
\Bibitem{BizBorMam14} \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 \mathnet{http://mi.mathnet.ru/rcd163} \crossref{https://doi.org/10.1134/S1560354714030095} \mathscinet{http://www.ams.org/mathscinet-getitem?mr=3215697} \zmath{https://zbmath.org/?q=an:1309.70020} \isi{http://gateway.isiknowledge.com/gateway/Gateway.cgi?GWVersion=2&SrcApp=PARTNER_APP&SrcAuth=LinksAMR&DestLinkType=FullRecord&DestApp=ALL_WOS&KeyUT=000337051600009} 

• http://mi.mathnet.ru/eng/rcd163
• http://mi.mathnet.ru/eng/rcd/v19/i3/p415

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Citing articles on Google Scholar: Russian citations, English citations
Related articles on Google Scholar: Russian articles, English articles

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
2. Andrey V. Tsiganov, “Killing Tensors with Nonvanishing Haantjes Torsion and Integrable Systems”, Regul. Chaotic Dyn., 20:4 (2015), 463–475
3. I. A. Bizyaev, A. V. Borisov, I. S. Mamaev, “Hamiltonization of elementary nonholonomic systems”, Russ. J. Math. Phys., 22:4 (2015), 444–453
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
5. A. V. Tsiganov, “Two integrable systems with integrals of motion of degree four”, Theoret. and Math. Phys., 186:3 (2016), 383–394
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
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
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
9. Shengda Hu, Manuele Santoprete, “Suslov Problem with the Clebsch–Tisserand Potential”, Regul. Chaotic Dyn., 23:2 (2018), 193–211