Abstract:
We generalize, to some extent, the results on integrable geodesic flows on two dimensional manifolds with a quartic first integral in the framework laid down by Selivanova and Hadeler. The local structure is first determined by a direct integration of the differential system which expresses the conservation of the quartic observable and is seen to involve a finite number of parameters. The global structure is studied in some detail and leads to a class of models on the manifolds S2, H2 or R2. As special cases we recover Kovalevskaya’s integrable system and a generalization of it due to Goryachev.
\Bibitem{Val13}
\by Galliano Valent
\paper On a Class of Integrable Systems with a Quartic First Integral
\jour Regul. Chaotic Dyn.
\yr 2013
\vol 18
\issue 4
\pages 394--424
\mathnet{http://mi.mathnet.ru/rcd120}
\crossref{https://doi.org/10.1134/S1560354713040060}
\mathscinet{http://mathscinet.ams.org/mathscinet-getitem?mr=3090209}
\zmath{https://zbmath.org/?q=an:1274.70024}
\isi{https://gateway.webofknowledge.com/gateway/Gateway.cgi?GWVersion=2&SrcApp=Publons&SrcAuth=Publons_CEL&DestLinkType=FullRecord&DestApp=WOS_CPL&KeyUT=000322878100006}
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This publication is cited in the following 10 articles:
Szuminski W. Maciejewski A.J., “Comment on “on the Integrability of 2D Hamiltonian Systems With Variable Gaussian Curvature” By a. a. Elmandouh”, Nonlinear Dyn., 104:2 (2021), 1443–1450
W. Szuminski, “Integrability analysis of natural Hamiltonian systems in curved spaces”, Commun. Nonlinear Sci. Numer. Simul., 64 (2018), 246–255
Andrey V. Tsiganov, “Bäcklund Transformations for the Nonholonomic Veselova System”, Regul. Chaotic Dyn., 22:2 (2017), 163–179
Andrey V. Tsiganov, “Integrable Discretization and Deformation of the Nonholonomic Chaplygin Ball”, Regul. Chaotic Dyn., 22:4 (2017), 353–367
I. A. Bizyaev, A. V. Borisov, I. S. Mamaev, “Generalizations of the Kovalevskaya case and quaternions”, Proc. Steklov Inst. Math., 295 (2016), 33–44
Galliano Valent, “Global Structure and Geodesics for Koenigs Superintegrable Systems”, Regul. Chaotic Dyn., 21:5 (2016), 477–509
A. V. Tsyganov, “Razdelenie peremennykh dlya odnogo obobscheniya sistemy Chaplygina na sfere”, Nelineinaya dinam., 11:1 (2015), 179–185
A. V. Tsiganov, “On the Chaplygin system on the sphere with velocity dependent potential”, J. Geom. Phys., 92 (2015), 94–99
G. Valent, Ch. Duval, V. Shevchishin, “Explicit metrics for a class of two-dimensional cubically superintegrable systems”, J. Geom. Phys., 87 (2015), 461–481
A. Galajinsky, O. Lechtenfeld, “On two-dimensional integrable models with a cubic or quartic integral of motion”, J. High Energy Phys., 2013, no. 9, 113
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