Bou-Rabee, Nawaf M. and Marsden, Jerrold E. and Romero, Louis A. (2005) A geometric treatment of Jellett’s egg. ZAMM - Journal of Applied Mathematics, 85 (9). pp. 618-642. ISSN 0044-2267 http://resolver.caltech.edu/CaltechAUTHORS:20100715-101709674
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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.
|Additional Information:||© 2005 WILEY-VCH Verlag. Received 22 April 2004, accepted 23 November 2004. Published online 10 June 2005. We wish to acknowledge Darryl Holm, Andy Ruina, Demetri Spanos, and Ahmed Bou-Rabee for helpful remarks. Bou-Rabee’s research was supported by the US DOE Computational Science Graduate Fellowship through grant DE-FG02-97ER25308; Marsden’s research partially supported by the National Science Foundation through NSF grant DMS-0204474; and Romero’s research was supported by Sandia National Laboratories. Sandia is a multiprogram laboratory operated by Sandia Corporation, a Lockheed Martin Company, for the United States Department of Energy under contract DE-AC04-94AL85000. The U.S. Government retains a nonexclusive, royalty-free license to publish or reproduce the published form of this contribution, or allowothers to do so, for U.S. Government purposes. Copyright is owned by ZAMM to the extent not limited by these rights.|
|Subject Keywords:||rising egg; dissipation induced instability; Jellett momentum map; adiabatic invariants|
|Classification Code:||MSC (2000) 70E18, 70F40, 34D23, 34E13, 37J15, 37M05, 37N05|
|Official Citation:||A geometric treatment of Jellett's egg (p 618-642) N.M. Bou-Rabee, J.E. Marsden, L.A. Romero Published Online: Jun 7 2005 8:16AM DOI: 10.1002/zamm.200410207|
|Usage Policy:||No commercial reproduction, distribution, display or performance rights in this work are provided.|
|Deposited By:||Ruth Sustaita|
|Deposited On:||15 Jul 2010 18:54|
|Last Modified:||26 Dec 2012 12:14|
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