A geometric treatment of Jellett's egg

Bou-Rabee, N. M., J. E. Marsden, and L. N. Romero

Z. Angew. Math. Mech. (ZAMM) 85, (2005), 618-642

Abstract:

This paper explains and gives a global analysis of the ``rising egg'' phenomenon. The main tools that are used in this analysis are derived from the theory of dissipation-induced instabilities, adiabatic invariants, and LaSalle's invariance principle. The analysis is done within the framework of a specific model of the egg as a prolate spheroid, with its equations of motion derived from Newtonian mechanics.

The paper begins by considering the linear and nonlinear stability of the non-risen and risen states of the spheroid corresponding to the initial and final state of the rising egg phenomenon. The asymptotic state of the spheroid is determined by an adiabatic momentum invariant. Because the symmetry associated with this adiabatic invariant coincides with the symmetry associated with the Jellett invariant in the tippe top, we call this quantity the Jellett momentum map.

Linear theory shows that the spectral stability of the non-risen state is determined by a cubic polynomial. The spectral stability of the risen state is governed by the modified Maxwell-Bloch equations--a normal form that appears in the problem of tippe top inversion and that was studied previously by the authors. A generalization of the energy-momentum method that includes adiabatic momentum invariants provides explicit criteria for the existence of an orbit connecting these states. In particular, it is shown that if the risen state is stable, the spheroid rises all the way.