Stability of Generic Equilibria of the 2n Dimensional Free Rigid Body Using the Energy-Casimir Method

http://hdl.handle.net/10150/194821
Title:
Stability of Generic Equilibria of the 2n Dimensional Free Rigid Body Using the Energy-Casimir Method
Author:
Issue Date:
2006
Publisher:
The University of Arizona.
Rights:
Copyright © is held by the author. Digital access to this material is made possible by the University Libraries, University of Arizona. Further transmission, reproduction or presentation (such as public display or performance) of protected items is prohibited except with permission of the author.
Abstract:
The rigid body has been one of the most noteworthy applications of Newtonian mechanics. Applying the principles of classical mechanics to the rigid body is by no means routine. The equations of motion, though discovered two hundred and fifty years ago by Euler, have remained quite elusive since their introduction. Understanding the rigid body has required the applications of concepts from integrable systems, algebraic geometry, Lie groups, representation theory, and symplectic geometry to name a few. Moreover, several important developments in these fields have in fact originated with the study of the rigid body and subsequently have grown into general theories with much wider applications.In this work, we study the stability of equilibria of non-degenerate orbits of the generalized rigid body. The energy-Casimir method introduced by V.I. Arnold in 1966 allows us to prove stability of certain non-degenerate equilibria of systems on Lie groups. Applied to the three dimensional rigid body, it recovers the classical Euler stability theorem [12]: rotations around the longest and shortest principal moments of inertia are stable equilibria. This method has not been applied to the analysis of rigid body dynamics beyond dimension n = 3. Furthermore, no conditions for the stability of equilibria are known at all beyond n = 4, in which case the conditions are not of the elegant longest/shortest type [10].Utilizing the rich geometric structures of the symmetry group G = SO(2n), we obtain stability results for generic equilibria of the even dimensional free rigid body. After obtaining a general expression for the generic equilibria, we apply the energy-Casimir method and find that indeed the classical longest/shortest conditions on the entries of the inertia matrix are suffcient to prove stability of generic equilibria for the generalized rigid body in even dimensions.
Type:
text; Electronic Dissertation
Keywords:
rigid body; symplectic geometry; Lie algebra; energy-Casimir
Degree Name:
PhD
Degree Level:
doctoral
Degree Program:
Degree Grantor:
University of Arizona
Flaschka, Hermann
Committee Chair:
Flaschka, Hermann

DC FieldValue Language
dc.language.isoENen_US
dc.titleStability of Generic Equilibria of the 2n Dimensional Free Rigid Body Using the Energy-Casimir Methoden_US
dc.date.issued2006en_US
dc.publisherThe University of Arizona.en_US
dc.rightsCopyright © is held by the author. Digital access to this material is made possible by the University Libraries, University of Arizona. Further transmission, reproduction or presentation (such as public display or performance) of protected items is prohibited except with permission of the author.en_US
dc.description.abstractThe rigid body has been one of the most noteworthy applications of Newtonian mechanics. Applying the principles of classical mechanics to the rigid body is by no means routine. The equations of motion, though discovered two hundred and fifty years ago by Euler, have remained quite elusive since their introduction. Understanding the rigid body has required the applications of concepts from integrable systems, algebraic geometry, Lie groups, representation theory, and symplectic geometry to name a few. Moreover, several important developments in these fields have in fact originated with the study of the rigid body and subsequently have grown into general theories with much wider applications.In this work, we study the stability of equilibria of non-degenerate orbits of the generalized rigid body. The energy-Casimir method introduced by V.I. Arnold in 1966 allows us to prove stability of certain non-degenerate equilibria of systems on Lie groups. Applied to the three dimensional rigid body, it recovers the classical Euler stability theorem [12]: rotations around the longest and shortest principal moments of inertia are stable equilibria. This method has not been applied to the analysis of rigid body dynamics beyond dimension n = 3. Furthermore, no conditions for the stability of equilibria are known at all beyond n = 4, in which case the conditions are not of the elegant longest/shortest type [10].Utilizing the rich geometric structures of the symmetry group G = SO(2n), we obtain stability results for generic equilibria of the even dimensional free rigid body. After obtaining a general expression for the generic equilibria, we apply the energy-Casimir method and find that indeed the classical longest/shortest conditions on the entries of the inertia matrix are suffcient to prove stability of generic equilibria for the generalized rigid body in even dimensions.en_US
dc.typetexten_US
dc.typeElectronic Dissertationen_US
dc.subjectrigid bodyen_US
dc.subjectsymplectic geometryen_US
dc.subjectLie algebraen_US
dc.subjectenergy-Casimiren_US
thesis.degree.namePhDen_US
thesis.degree.leveldoctoralen_US
thesis.degree.disciplineMathematicsen_US