INFORMATION TO USERS This manuscript has been reproduced from the microfilm master. UMI films the text directly from the original or copy submitted. Thus, some thesis and dissertation copies are in typewriter face, while others may be from any type o f computer printer. The quality of this reproduction is dependent upon the quality of the copy submitted. Broken or indistinct print, colored or poor quality illustrations and photographs, print bleedthrough, substandard margins, and improper alignment can adversely affect reproduction. In the unlikely event that the author did not send UMI a complete manuscript and there are missing pages, these will be noted. Also, if unauthorized copyright material had to be removed, a note will indicate the deletion. Oversize materials (e.g., maps, drawings, charts) are reproduced by sectioning the original, beginning at the upper left-hand comer and continuing from left to right in equal sections with small overlaps. Each original is also photographed in one exposure and is included in reduced form at the back o f the book. Photographs included in the original manuscript have been reproduced xerographically in this copy. Ifrgher quality 6” x 9” black and white photographic prints are available for any photographs or illustrations appearing in this copy for an additional charge. Contact UMI directly to order. UMI A Bell & Howell Infomiation Company 300 North Zeeb Road, Ann Aibor MI 48106-1346 USA 313/761-4700 800/521-0600 INTEGRABILITY AND STABILITY OF NONHOLONOMIC SYSTEMS DISSERTATION Presented in Partial Fulfillment of the Requirements for the Degree Doctor of Philosophy in the Graduate School of The Ohio State University By Dmitry V. Zenkov, M.S. ***** The Ohio State University 1998 Dissertation Committee: . , , Approved by Professor Anthony Bloch, Advisor Professor Alexander Dynin ~ , .... ' Advisor Professor Andrzej Derdzinski Department of Mathematis UMI Number: 9834105 UMI Microform 9834105 Copyright 1998, by UMI Company. All rights reserved. This microform edition is protected against unauthorized copying under Title 17, United States Code. UMI 300 North Zeeh Road Ann Arbor, MI 48103 ABSTRACT In this thesis, methods of geometric mechanics are used to study the integrability and stability of nonholonomic systems (that is, mechanical systems with noninte- grable constraints such as rolling constraints). In the absence of external dissipation, such systems conserve energy, but nonetheless can exhibit neutrally stable and asymp­ totically stable, as well as linearly unstable relative equilibria. Unlike Hamiltonian dynamics, symmetries do not always lead to conservation laws as in the classical Noether theorem, but rather to a dynamic momentum equation. This momentum equation has the structure of a parallel transport equation for momentum corrected by additional terms and plays an important role in both integrability and stability analyses. For a number of systems, the momentum equation is pure transport. For such systems we show that the relative equilibria cannot be asymptotically stable and we find the energy-momentum conditions for stability of these equilibria. Often in this case of a pure transport momentum equation, the nonholonomic system is integrable. .A.mong these systems are the Routh problem and the rolling disk. We show that the invariant manifolds of the Routh problem are tori filled out with quasi-periodic motions. However, the way these tori are embedded in the phase space is quite different from that of the Liouville tori of integrable Hamiltonian systems. To carry out the stability analysis in the general non-pure transport case, we use a generalization of the energy-momentum method for holonomie systems combined with the Lyapunov-Malkin theorem and the center manifold theorem. We develop a 11 new energy-based approach, which allows one to find asymptotically stable equilibria. While this approach is consistent with the energy-momentum method for holonomie systems, it extends it in substantial ways. The theory is illustrated with several examples, including the rolling disk, the roller racer, and the rattleback top. Ill To my parents IV ACKNOWLEDGMENTS I would like to thank my advisor, Tony Bloch, for taking the time to have numerous discussions with me and for his encouragement, support and patience. This thesis could never have been done without his enthusiasm. I am grateful to Jerry Marsden for his hospitality during my visit to Caltech, the interesting discussions, and his help. I learned a lot during this visit. It was my pleasure to work with Tony Bloch and Jerry Marsden on the iionholo- nomic stability project. I am thankful to Andrzej Derdzinski and Alexander Dynin for their discussions with me and to Greg Forest for his interest in my research and introducing me in 1993 to Tonv Bloch. VITA October 12. 1962 ................................ Bom—Moscow, Russia 1986 ..................................................... M.S. Mechanics, Moscow State University 1986-1989 ............................................ Graduate School Department of Mechanics and Mathematics Moscow State University 1990-1993 .............................................Assistant Professor, Moscow Technical State University 1993-present ........................................Graduate Teaching Associate, Department of Mathematics. The Ohio State University PUBLICATIONS Research Publications 1 V.V. Kozlov and D.V. Zenkov, On Geometric Poinsot Interpretation for an n- dimensional Rigid Body, Tr. Semin. Vectom. Tenzom. Anal, 23 (1988). 202-204 2 D.V. Zenkov, On Asymptotic Stability of Periodic Motions in Nonholonomic Mechanics, Vestnik Moskov. Univ. Ser. I Math. Mekh., 3 (1989), 46-51 3 D.V. Zenkov, On the Routh Problem, Vestnik Moskov. Univ. Ser. I Math. Mekh., 3 (1991), 87-89 4 D.V. Zenkov, On the Problem of a Sphere Rolling over a Surface of Revolution, Vestnik Moskov. Univ. Ser. I Math. MeAh., 4 (1991), 94-96 VI 5 D.V. Zenkov, The Geometry of the Routh Problem, J. Nonlinear Sci. 5 (1995), 503-519 6 D.V. Zenkov, A.M. Bloch, and J.E. Marsden, The Energy-Momentum Method for Stability of Nonholonomic Systems, (to appear in Dynamics and Stability of Systems) FIELDS OF STUDY Major Field: Mathematics Specialization: Applied Mathematics Vll TABLE OF CONTENTS Abstract ii Acknowledgments v V ita vi List of Figures x Introduction 1 1 Overview of Nonholonomic Mechanics and Stability Theory 5 1.1 The Hamilton Principle ............................................................................... 5 1.2 The Equations of Motion of Nonholonomic Systems with Symmetries 6 1.2.1 The Lagrange-d’Alembert Principle ............................................. 6 1.2.2 Sym m etries ...................................................................................... 9 1.3 Examples of Nonholonomic System s ......................................................... 9 1.3.1 The Rolling D isk ............................................................................. 10 1.3.2 The Routh Problem ....................................................................... 11 1.3.3 A Mathematical Exam ple ............................................................. 11 1.3.4 The Roller R a c e r ............................................................................. 13 1.3.5 The Rattleback ................................................................................ 14 1.4 The Geometry of Nonholonomic Systems with Symmetry .................. 16 1.5 Lyapunov Stability T heory ........................................................................ 25 1.5.1 Main Definitions and T h eo rem s .................................................... 25 1.5.2 Center Manifold Theory in Stability Analysis .......................... 27 1.6 The Energy-Momentum Method for Holonomie Systems...................... 30 1.7 Invariant Manifolds ..................................................................................... 32 2 The Routh Problem 35 2.1 Integrability of the Routh Problem ........................................................ 35 2.2 Stability of Stationary Periodic Motions of the Routh Problem .... 40 2.3 Integral Manifolds of the Routh P ro b lem ............................................... 47 vm 3 An Energy-Momentum Method for Stability of Nonholonomic Systems 55 3.1 The Pure Transport Case ......................................................................... 57 3.1.1 The Nonholonomic Energy-Momentum M ethod ......................... 58 3.1.2 The Rolling D i s c .............................................................................. 61 3.2 The Non-Pure Transport Case .................................................................. 63 3.2.1 The Mathematical E x am p le ........................................................... 63 3.2.2 The Nonholonomic Energy-Momentum M ethod ........................ 67 3.2.3 The Roller R a c e r .............................................................................. 75 3.3 Nonlinear Stability by the Lyapunov-Malkin M ethod .................. 77 3.3.1 The Lyapunov-Malkin and the Energy-Momentum Methods 79 3.3.2 The R attleback ................................................................................. 82 Conclusion 86 Bibliography 87 IX LIST OF FIGURES 1.1 The geometry for the rolling disk ................................................................ 10 1.2 The geometry for the roller racer .................................................................. 14 1.3 The rattleback...............................................................................................