Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)


SIGMA 3 (2007), 041, 14 pages      nlin.SI/0703016      http://dx.doi.org/10.3842/SIGMA.2007.041
Contribution to the Vadim Kuznetsov Memorial Issue

Phase Space of Rolling Solutions of the Tippe Top

S. Torkel Glad a, Daniel Petersson a and Stefan Rauch-Wojciechowski b
a) Dept. of Electrical Engineering, Linköpings Universitet SE-581 83 Linköping, Sweden
b) Department of Mathematics, Linköpings Universitet, SE-581 83 Linköping, Sweden

Received September 15, 2006, in final form February 05, 2007; Published online March 09, 2007

Abstract
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 (θ,φ,ψ) these integrals give separation equations that have the same structure as the equations of the Lagrange top. It makes it possible to describe the whole space of solutions by representing them in the space of parameters (D,λ,E) being constant values of the integrals of motion.

Key words: nonholonomic dynamics; rigid body; rolling sphere; tippe top; integrals of motion.

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