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Phase space of rolling solutions of the tippe top. (English) Zbl 1131.70003

Summary: Equations of motion of an axially symmetric sphere rolling and sliding on a plane are usually taken as model of the tippe top. We study these equations in the nonsliding regime both in the vector notation and in the Euler angle variables when they admit three integrals of motion that are linear and quadratic in momenta. In the Euler angle variables \(\theta,\phi,\psi\) these integrals give separation of equations that have the same structure as the equations of Lagrange top. It makes it possible to describe the whole space of solutions by representing them in the space of parameters \(D,\lambda,E\) being constant values of the integrals of motion.

MSC:

70E18 Motion of a rigid body in contact with a solid surface
70E40 Integrable cases of motion in rigid body dynamics
70F25 Nonholonomic systems related to the dynamics of a system of particles
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