Abstract:
We consider an analogue of the 4th Appelrot class of motions of the Kowalevski top for the case of two constant force fields. The trajectories of this family fill a four-dimensional surface O in the six-dimensional phase space. The constants of the three first integrals in involution restricted to this surface fill one of the sheets of the bifurcation diagram in R3. We point out a pair of partial integrals to obtain explicit parametric equations of this sheet. The induced system on O is shown to be Hamiltonian with two degrees of freedom having a thin set of points where the induced symplectic structure degenerates. The region of existence of motions in terms of the integral constants is found. We provide the separation of variables on O and algebraic formulae for the initial phase variables
Keywords:
Kowalevski top, double field, Appelrot classes, separation of variables.
\Bibitem{Kha07}
\by M. P. Kharlamov
\paper Separation of Variables in the Generalized 4th Appelrot Class
\jour Regul. Chaotic Dyn.
\yr 2007
\vol 12
\issue 3
\pages 267--280
\mathnet{http://mi.mathnet.ru/rcd624}
\crossref{https://doi.org/10.1134/S1560354707030021}
\mathscinet{http://mathscinet.ams.org/mathscinet-getitem?mr=2350325}
\zmath{https://zbmath.org/?q=an:1229.70012}
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https://www.mathnet.ru/eng/rcd624
https://www.mathnet.ru/eng/rcd/v12/i3/p267
This publication is cited in the following 9 articles:
M.P. Kharlamov, “Phase topology of one system with separated variables and singularities of the symplectic structure”, Journal of Geometry and Physics, 87 (2015), 248
M. P. Kharlamov, P. E. Ryabov, “Topological atlas of the Kovalevskaya top in a double field”, J. Math. Sci., 223:6 (2017), 775–809
Mikhail P. Kharlamov, “Extensions of the Appelrot Classes for the Generalized
Gyrostat in a Double Force Field”, Regul. Chaotic Dyn., 19:2 (2014), 226–244
P. E. Ryabov, “The phase topology of a special case of Goryachev integrability in rigid body dynamics”, Sb. Math., 205:7 (2014), 1024–1044
P. E. Ryabov, “Phase topology of one irreducible integrable problem in the dynamics
of a rigid body”, Theoret. and Math. Phys., 176:2 (2013), 1000–1015
P. E. Ryabov, M. P. Kharlamov, “Classification of singularities in the problem of motion of the Kovalevskaya top in a double force field”, Sb. Math., 203:2 (2012), 257–287
P. E. Ryabov, “Explicit integration and topology of D.N. Goryachev case”, Dokl. Math., 84:1 (2011), 502
Mikhail P. Kharlamov, “Bifurcation diagrams and critical subsystems of the Kowalevski gyrostat in two constant fields”, Hiroshima Math. J., 39:3 (2009)
M. P. Kharlamov, “Separation of variables in the generalized 4th Appelrot class. II. Real solutions”, Regul. Chaotic Dyn., 14:6 (2009), 621–634
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