Abstract:
In an earlier paper Orel computed the trajectory (or orbital) invariant for the Goryachev–Chaplygin problem. She noted, however, that the investigation of this invariant is beset with insurmountable analytic difficulties. The present paper completes the construction of the trajectory invariant for the Goryachev–Chaplygin problem, as well as for the Lagrange problem, by the methods of computer analysis. Thus the orbital classification question as been solved for these problems. In the paper we also formulate conjectures relating to the Lagrange case, which can serve as a basis for further investigations in this area. The computational algorithms themselves are due to Takahashi.
Citation:
O. E. Orel, S. Takahashi, “Orbital classification of the integrable problems of Lagrange and Goryachev–Chaplygin by the methods of computer analysis”, Sb. Math., 187:1 (1996), 93–110
\Bibitem{OreTak96}
\by O.~E.~Orel, S.~Takahashi
\paper Orbital classification of the~integrable problems of Lagrange and Goryachev--Chaplygin by the~methods of computer analysis
\jour Sb. Math.
\yr 1996
\vol 187
\issue 1
\pages 93--110
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Linking options:
https://www.mathnet.ru/eng/sm102
https://doi.org/10.1070/SM1996v187n01ABEH000102
https://www.mathnet.ru/eng/sm/v187/i1/p95
This publication is cited in the following 9 articles:
V. V. Vedyushkina (Fokicheva), A. T. Fomenko, “Integrable topological billiards and equivalent dynamical systems”, Izv. Math., 81:4 (2017), 688–733
Fokicheva V.V., Fomenko A.T., “Billiard Systems as the Models For the Rigid Body Dynamics”, Advances in Dynamical Systems and Control, Studies in Systems Decision and Control, 69, eds. Sadovnichiy V., Zgurovsky M., Springer Int Publishing Ag, 2016, 13–33
Fomenko A.T. Nikolaenko S.S., “The Chaplygin Case in Dynamics of a Rigid Body in Fluid Is Orbitally Equivalent To the Euler Case in Rigid Body Dynamics and To the Jacobi Problem About Geodesics on the Ellipsoid”, J. Geom. Phys., 87 (2015), 115–133
D. Bertrand, “Multiplicity and vanishing lemmas for differential and q-difference equations in the Siegel–Shidlovsky theory”, J Math Sci, 2012
E. O. Kantonistova, “Integer lattices of the action variables for the generalized Lagrange case”, Moscow University Mathematics Bulletin, 67:1 (2012), 36–40
Korovina, NV, “The trajectory equivalence of two classical problems in rigid body dynamics”, Doklady Mathematics, 62:3 (2000), 345
A. V. Bolsinov, A. T. Fomenko, Integrable Geodesic Flows on Two-Dimensional Surfaces, 2000, 287
O. E. Orel, “A criterion for orbital equivalence of integrable Hamiltonian systems in the vicinity of elliptic orbits. An orbital invariant in the Lagrange problem”, Sb. Math., 188:7 (1997), 1085–1105
A. V. Bolsinov, “Fomenko invariants in the theory of integrable Hamiltonian systems”, Russian Math. Surveys, 52:5 (1997), 997–1015
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