Аннотация:
We consider the motion of a material point on the surface of a sphere in the field of 2n+1 identical Hooke centers (singularities with elastic potential) lying on a great circle. Our main result is that this system is superintegrable. The property of superintegrability for this system has been conjectured by us in [1], where the structure of a superintegral of arbitrarily high odd degree in momemnta was outlined. We also indicate an isomorphism between this system and the one-dimensional N-particle system discussed in the recent paper [2] and show that for the latter system an analogous superintegral can be constructed.
Ключевые слова:
superintegrable systems, systems with a potential, Hooke center.
Поступила в редакцию: 21.10.2009 Принята в печать: 16.11.2009
Образец цитирования:
A. V. Borisov, A. A. Kilin, I. S. Mamaev, “Superintegrable system on a sphere with the integral of higher degree”, Regul. Chaotic Dyn., 14:6 (2009), 615–620
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\by A. V. Borisov, A. A. Kilin, I. S. Mamaev
\paper Superintegrable system on a sphere with the integral of higher degree
\jour Regul. Chaotic Dyn.
\yr 2009
\vol 14
\issue 6
\pages 615--620
\mathnet{http://mi.mathnet.ru/rcd1002}
\crossref{https://doi.org/10.1134/S156035470906001X}
\mathscinet{http://mathscinet.ams.org/mathscinet-getitem?mr=2591863}
\zmath{https://zbmath.org/?q=an:1229.70052}
Образцы ссылок на эту страницу:
https://www.mathnet.ru/rus/rcd1002
https://www.mathnet.ru/rus/rcd/v14/i6/p615
Эта публикация цитируется в следующих 15 статьяx:
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
Gonera C. Gonera J., “New Superintegrable Models on Spaces of Constant Curvature”, Ann. Phys., 413 (2020), 168052
G Gubbiotti, D Latini, “A multiple scales approach to maximal superintegrability”, J. Phys. A: Math. Theor., 51:28 (2018), 285201
Claudia Maria Chanu, Giovanni Rastelli, “Extended Hamiltonians and shift, ladder functions and operators”, Annals of Physics, 386 (2017), 254
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
Claudia Maria Chanu, Luca Degiovanni, Giovanni Rastelli, “Extended Hamiltonians, Coupling-Constant Metamorphosis and the Post–Winternitz System”, SIGMA, 11 (2015), 094, 9 pp.
Ángel Ballesteros, Alfonso Blasco, Francisco J Herranz, Fabio Musso, “An integrable Hénon–Heiles system on the sphere and the hyperbolic plane”, Nonlinearity, 28:11 (2015), 3789
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
Claudia Maria Chanu, Luca Degiovanni, Giovanni Rastelli, “The Tremblay-Turbiner-Winternitz system as extended Hamiltonian”, Journal of Mathematical Physics, 55:12 (2014)
Valery V. Kozlov, “Remarks on Integrable Systems”, Regul. Chaotic Dyn., 19:2 (2014), 145–161
Andrey V. Tsiganov, “Superintegrable Stäckel systems on the plane: elliptic and parabolic coordinates”, SIGMA, 8 (2012), 031, 9 pp.
Claudia Chanu, Luca Degiovanni, Giovanni Rastelli, “First Integrals of Extended Hamiltonians in $n+1$ Dimensions Generated by Powers of an Operator”, SIGMA, 7 (2011), 038, 12 pp.
A.J. Maciejewski, M. Przybylska, A.V. Tsiganov, “On algebraic construction of certain integrable and super-integrable systems”, Physica D: Nonlinear Phenomena, 240:18 (2011), 1426
Claudia Chanu, Luca Degiovanni, Giovanni Rastelli, “Three and Four-body Systems in One Dimension: Integrability, Superintegrability and Discrete Symmetries”, Regul. Chaotic Dyn., 16:5 (2011), 496–503
Andrzej J Maciejewski, Maria Przybylska, Haruo Yoshida, “Necessary conditions for classical super-integrability of a certain family of potentials in constant curvature spaces”, J. Phys. A: Math. Theor., 43:38 (2010), 382001
Цитирование в формате AMSBIB
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