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Rolling of a non-homogeneous ball over a sphere without slipping and twisting. (English) Zbl 1229.37081

Summary: Consider the problem of rolling a dynamically asymmetric balanced ball (the Chaplygin ball) over a sphere. Suppose that the contact point has zero velocity and the projection of the angular velocity to the normal vector of the sphere equals zero. This model of rolling differs from the classical one. It can be realized, in some approximation, if the ball is rubber coated and the sphere is absolutely rough. Recently, J. Koiller and K. Ehlers pointed out the measure and the Hamiltonian structure for this problem. Using this structure we construct an isomorphism between this problem and the problem of the motion of a point on a sphere in some potential field. The integrable cases are found.

MSC:

37J60 Nonholonomic dynamical systems
37N05 Dynamical systems in classical and celestial mechanics
76M23 Vortex methods applied to problems in fluid mechanics
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[1] Ehlers, K., Koiller, J., Montgomery, R. and Rios, P.M., Nonholonomic Systems via Moving Frames: Cartan Equivalence and Chaplygin Hamiltonization, in Marsden, J. and Ratiu, T., Eds., The Breadth of Symplectic and Poisson Geometry, vol. 132 of Progr. Math, Boston: Birkhäuser, 2005, pp. 75–120. · Zbl 1095.70007
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