Analysis of the phase space, asymptotic behavior and stability for heavy symmetric top and tippe top

University dissertation from Matematiska institutionen

Abstract: In this thesis we analyze the phase of the heavy symmetric top and the tippe top. These tops are two examples of physical systems for which the usefulness of integrals of motion and invariant manifolds, in phase space picture analysis, can be illustratedIn the case of the heavy symmetric top, simplified proofs of stability of the vertical rotation have been perpetuated by successive textbooks during the last century. In these proofs correct perturbations of integrals of motion are missing. This may seem harmless since the deduced threshold value for stability is correct. However, perturbations of first integrals are essential in rigorous proofs of stability of motions for both tops.The tippe top is a toy that has the form of a truncated sphere equipped with a little peg. When spun fast on the spherical bottom its center of mass rises above its geometrical center and after a few seconds the top is spinning vertically on the peg. We study the tippe top through a sequence of embedded invariant manifolds to unveil the structure of the top's phase space. The last manifold, consisting of the asymptotic trajectories, is analyzed completely. We prove that trajectories in this manifold attract solutions in contact with the plane of support at all times and we give a complete description of their stability/instability properties for all admissible choices of model parameters and of the initial conditions.

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