Geodesic
Generalized Chaplygins transformation and explicit integration of a system with a spherical support
Russian journal of nonlinear dynamics, Tome 7 (2011) no. 2, pp. 313-338

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We consider the problem of explicit integration and bifurcation analysis for two systems of nonholonomic mechanics. The first one is the Chaplygin's problem on no-slip rolling of a balanced dynamically non-symmetrical ball on a horizontal plane. The second problem is on the motion of rigid body in a spherical support. We explicitly integrate this problem by generalizing the transformation which Chaplygin applied to the integration of the problem of the rolling ball at a non-zero constant of areas. We consider the geometric interpretation of this transformation from the viewpoint of a trajectory isomorphism between two systems at different levels of the energy integral. Generalization of this transformation for the case of dynamics in a spherical support allows us to integrate the equations of motion explicitly in quadratures and, in addition, to indicate periodic solutions and analyze their stability. We also show that adding a gyrostat does not lead to the loss of integrability.
Keywords: nonholonomic mechanics, spherical support, Chaplygin ball, explicit integration, bifurcation analysis.
Mots-clés : isomorphism 
@article{ND_2011_7_2_a7,
     author = {A. V. Borisov and A. A. Kilin and I. S. Mamaev},
     title = {Generalized {Chaplygins} transformation and explicit integration of a system with a spherical support},
     journal = {Russian journal of nonlinear dynamics},
     pages = {313--338},
     publisher = {mathdoc},
     volume = {7},
     number = {2},
     year = {2011},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/ND_2011_7_2_a7/}
}
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A. V. Borisov; A. A. Kilin; I. S. Mamaev. Generalized Chaplygins transformation and explicit integration of a system with a spherical support. Russian journal of nonlinear dynamics, Tome 7 (2011) no. 2, pp. 313-338. http://geodesic.mathdoc.fr/item/ND_2011_7_2_a7/