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Symmetry, Integrability and Geometry: Methods and Applications, 2015, Volume 11, 088, 51 pp.
DOI: https://doi.org/10.3842/SIGMA.2015.088
(Mi sigma1069)
 

This article is cited in 4 scientific papers (total in 4 papers)

Examples of Complete Solvability of 2D Classical Superintegrable Systems

Yuxuan Chena, Ernie G. Kalninsb, Qiushi Lia, Willard Miller Jr.a

a School of Mathematics, University of Minnesota, Minneapolis, Minnesota, 55455, USA
b Department of Mathematics, University of Waikato, Hamilton, New Zealand
References:
Abstract: Classical (maximal) superintegrable systems in $n$ dimensions are Hamiltonian systems with $2n-1$ independent constants of the motion, globally defined, the maximum number possible. They are very special because they can be solved algebraically. In this paper we show explicitly, mostly through examples of $2$nd order superintegrable systems in $2$ dimensions, how the trajectories can be determined in detail using rather elementary algebraic, geometric and analytic methods applied to the closed quadratic algebra of symmetries of the system, without resorting to separation of variables techniques or trying to integrate Hamilton's equations. We treat a family of $2$nd order degenerate systems: oscillator analogies on Darboux, nonzero constant curvature, and flat spaces, related to one another via contractions, and obeying Kepler's laws. Then we treat two $2$nd order nondegenerate systems, an analogy of a caged Coulomb problem on the $2$-sphere and its contraction to a Euclidean space caged Coulomb problem. In all cases the symmetry algebra structure provides detailed information about the trajectories, some of which are rather complicated. An interesting example is the occurrence of “metronome orbits”, trajectories confined to an arc rather than a loop, which are indicated clearly from the structure equations but might be overlooked using more traditional methods. We also treat the Post–Winternitz system, an example of a classical $4$th order superintegrable system that cannot be solved using separation of variables. Finally we treat a superintegrable system, related to the addition theorem for elliptic functions, whose constants of the motion are only rational in the momenta. It is a system of special interest because its constants of the motion generate a closed polynomial algebra. This paper contains many new results but we have tried to present most of the materials in a fashion that is easily accessible to nonexperts, in order to provide entrée to superintegrablity theory.
Keywords: superintegrable systems; classical trajectories.
Received: May 5, 2015; in final form October 27, 2015; Published online November 3, 2015
Bibliographic databases:
Document Type: Article
MSC: 20C99; 20C35; 22E70
Language: English
Citation: Yuxuan Chen, Ernie G. Kalnins, Qiushi Li, Willard Miller Jr., “Examples of Complete Solvability of 2D Classical Superintegrable Systems”, SIGMA, 11 (2015), 088, 51 pp.
Citation in format AMSBIB
\Bibitem{CheKalLi15}
\by Yuxuan~Chen, Ernie~G.~Kalnins, Qiushi~Li, Willard~Miller Jr.
\paper Examples of Complete Solvability of 2D Classical Superintegrable Systems
\jour SIGMA
\yr 2015
\vol 11
\papernumber 088
\totalpages 51
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\crossref{https://doi.org/10.3842/SIGMA.2015.088}
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\scopus{https://www.scopus.com/record/display.url?origin=inward&eid=2-s2.0-84946552080}
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  • This publication is cited in the following 4 articles:
    Citing articles in Google Scholar: Russian citations, English citations
    Related articles in Google Scholar: Russian articles, English articles
    Symmetry, Integrability and Geometry: Methods and Applications
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