Abstract:
For the classical problem of motion of a rigid body about a fixed point with zero area integral, we present a family of solutions that are periodic in the absolute space. Such solutions are known as choreographies. The family includes the well-known Delone solutions (for the Kovalevskaya case), some particular solutions for the Goryachev–Chaplygin case, and the Steklov solution. The "genealogy" of solutions of the family naturally appearing from the energy continuation and their connection with the Staude rotations are considered. It is shown that if the integral of areas is zero, the solutions are periodic with respect to a coordinate frame that rotates uniformly about the vertical (relative choreographies).
Keywords:
rigid-body dynamics, periodic solutions, continuation by a parameter, bifurcation.
Citation:
A. V. Borisov, A. A. Kilin, I. S. Mamaev, “Absolute and Relative Choreographies in Rigid Body Dynamics”, Regul. Chaotic Dyn., 13:3 (2008), 204–220
\Bibitem{BorKilMam08}
\by A.~V.~Borisov, A.~A.~Kilin, I.~S.~Mamaev
\paper Absolute and Relative Choreographies in Rigid Body Dynamics
\jour Regul. Chaotic Dyn.
\yr 2008
\vol 13
\issue 3
\pages 204--220
\mathnet{http://mi.mathnet.ru/rcd571}
\crossref{https://doi.org/10.1134/S1560354708030064}
\mathscinet{http://mathscinet.ams.org/mathscinet-getitem?mr=2415374}
\zmath{https://zbmath.org/?q=an:1229.70009}
Linking options:
https://www.mathnet.ru/eng/rcd571
https://www.mathnet.ru/eng/rcd/v13/i3/p204
This publication is cited in the following 5 articles:
Borisov A. Mamaev I., “Rigid Body Dynamics”, Rigid Body Dynamics, de Gruyter Studies in Mathematical Physics, 52, Walter de Gruyter Gmbh, 2019, 1–520
Hamad M. Yehia, “New solvable problems in the dynamics of a rigid body about a fixed point in a potential field”, Mechanics Research Communications, 57 (2014), 44
Sebastián Ferrer, Francisco J. Molero, “Andoyer's variables and phases in the free rigid body”, Journal of Geometric Mechanics, 6:1 (2014), 25
A. V. Bolsinov, A. V. Borisov, I. S. Mamaev, “Topology and stability of integrable systems”, Russian Math. Surveys, 65:2 (2010), 259–318
A. V. Borisov, A. A. Kilin, I. S. Mamaev, “Chaos in a Restricted Problem of Rotation of a Rigid Body with a Fixed Point”, Regul. Chaotic Dyn., 13:3 (2008), 221–233
Citation in format AMSBIB
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