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.
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.
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
This publication is cited in the following 16 articles:
Andrey V. Tsiganov, “Rotations and Integrability”, Regul. Chaotic Dyn., 29:6 (2024), 913–930
Cezary Gonera, Joanna Gonera, Javier de Lucas, Wioletta Szczesek, Bartosz M. Zawora, “More on Superintegrable Models
on Spaces of Constant Curvature”, Regul. Chaotic Dyn., 27:5 (2022), 561–571
Latini D., Marquette I., Zhang Ya.-Zh., “Racah Algebra R(N) From Coalgebraic Structures and Chains of R(3) Substructures”, J. Phys. A-Math. Theor., 54:39 (2021), 395202
Latini D., Marquette I., Zhang Ya.-Zh., “Embedding of the Racah Algebra R(N) and Superintegrability”, Ann. Phys., 426 (2021), 168397
Gonera C., Gonera J., “New Superintegrable Models on Spaces of Constant Curvature”, Ann. Phys., 413 (2020), 168052
D. Latini, “Universal chain structure of quadratic algebras for superintegrable systems with coalgebra symmetry”, J. Phys. A-Math. Theor., 52:12 (2019), 125202
Shengda Hu, Manuele Santoprete, “Suslov Problem with the Clebsch–Tisserand Potential”, Regul. Chaotic Dyn., 23:2 (2018), 193–211
G. Gubbiotti, D. Latini, “A multiple scales approach to maximal superintegrability”, J. Phys. A-Math. Theor., 51:28 (2018), 285201
Galliano Valent, “Superintegrable Models on Riemannian Surfaces of Revolution with Integrals of any Integer Degree (I)”, Regul. Chaotic Dyn., 22:4 (2017), 319–352
A. V. Tsiganov, “Two integrable systems with integrals of motion of degree four”, Theoret. and Math. Phys., 186:3 (2016), 383–394
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
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
Andrey V. Tsiganov, “Killing Tensors with Nonvanishing Haantjes Torsion and Integrable Systems”, Regul. Chaotic Dyn., 20:4 (2015), 463–475
I. A. Bizyaev, A. V. Borisov, I. S. Mamaev, “Hamiltonization of elementary nonholonomic systems”, Russ. J. Math. Phys., 22:4 (2015), 444–453
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
I. A. Bizyaev, “Ob odnom obobschenii sistem tipa Kalodzhero”, Nelineinaya dinam., 10:2 (2014), 209–212
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