Geometric Mechanics - Part Ii: Rotating, Translating And Rolling (2nd Edition)

by Darryl D Holm

Overview

Featured in the recommended literature list for the International Society of Nonlinear Mathematical Physics: https://isnmp.de/Book-Reviews-and-Recommendations/See also GEOMETRIC MECHANICS — Part I: Dynamics and Symmetry (2nd Edition) This textbook introduces modern geometric mechanics to advanced undergraduates and beginning graduate students in mathematics, physics and engineering. In particular, it explains the dynamics of rotating, spinning and rolling rigid bodies from a geometric viewpoint by formulating their solutions as coadjoint motions generated by Lie groups. The only prerequisites are linear algebra, multivariable calculus and some familiarity with Euler-Lagrange variational principles and canonical Poisson brackets in classical mechanics at the beginning undergraduate level.The book uses familiar concrete examples to explain variational calculus on tangent spaces of Lie groups. Through these examples, the student develops skills in performing computational manipulations, starting from vectors and matrices, working through the theory of quaternions to understand rotations, then transferring these skills to the computation of more abstract adjoint and coadjoint motions, Lie-Poisson Hamiltonian formulations, momentum maps and finally dynamics with nonholonomic constraints.The organisation of the first edition has been preserved in the second edition. However, the substance of the text has been rewritten throughout to improve the flow and to enrich the development of the material. Many worked examples of adjoint and coadjoint actions of Lie groups on smooth manifolds have also been added and the enhanced coursework examples have been expanded. The second edition is ideal for classroom use, student projects and self-study.

Product Details

ISBN-13:	9781848167780
Publisher:	Imperial College Press
Publication date:	11/02/2011
Pages:	412
Product dimensions:	6.00(w) x 8.90(h) x 0.90(d)

Table of Contents

Preface xv

1 Galileo 1

1.1 Principle of Galilean relativity 2

1.2 Galilean transformations 3

1.2.1 Admissible force laws for an N-particle system 6

1.3 Subgroups of the Galilean transformations 8

1.3.1 Matrix representation of SE(3) 9

1.4 Lie group actions of SE(3) 11

1.5 Lie group actions of G(3) 12

1.5.1 Matrix representation of G(3) 14

1.6 Lie algebra of SE(3) 15

1.7 Lie algebra of G(3) 17

2 Newton, Lagrange, Hamilton and the rigid body 19

2.1 Newton 21

2.1.1 Newtonian form of free rigid rotation 21

2.1.2 Newtonian form of rigid-body motion 30

2.2 Lagrange 36

2.2.1 The principle of stationary action 36

2.3 Noether's theorem 39

2.3.1 Lie symmetries and conservation laws 39

2.3.2 Infinitesimal transformations of a Lie group 40

2.4 Lagrangian form of rigid-body motion 46

2.4.1 Hamilton-Pontryagin constrained variations 50

2.4.2 Manakov's formulation of the SO(n) rigid body 54

2.4.3 Matrix Euler-Poincaré equations 55

2.4.4 An isospectral eigenvalue problem for the SO(n) rigid body 56

2.4.5 Manakov's integration of the SO(n) rigid body 58

2.5 Hamilton 60

2.5.1 Hamiltonian form of rigid-body motion 62

2.5.2 Lie-Poisson Hamiltonian rigid-body dynamics 63

2.5.3 Lie-Poisson bracket 64

2.5.4 Nambu's R³ Poisson bracket 65

2.5.5 Clebsch variational principle for the rigid body 69

2.5.6 Rotating motion with potential energy 72

3 Quaterions 77

3.1 Operating with quaternions 78

3.1.1 Multiplying quaternions using Pauli matrices 79

3.1.2 Quaternionic conjugate 82

3.1.3 Decomposition of three-vectors 85

3.1.4 Alignment dynamics for Newton's second law 86

3.1.5 Quaternionic dynamics of Kepler's problem 90

3.2 Quaternionic conjugation 93

3.2.1 Cayley-Klein parameters 93

3.2.2 Pure quaternions, Pauli matrices and SU(2) 99

3.2.3 Tilde map: R³ su(2) so(3) 102

3.2.4 Dual of the tilde map: R^3* su(2)^* so(3)^* 103

3.2.5 Pauli matrices and Poincaré's sphere C² → S² 103

3.2.6 Poincaré's sphere and Hopf's fibration 105

3.2.7 Coquaternions 108

4 Adjoint and coadjoint actions 111

4.1 Cayley-Klein dynamics for the rigid body 112

4.1.1 Cayley-Klein parameters, rigid-body dynamics 112

4.1.2 Body angular frequency 113

4.1.3 Cayley-Klein parameters 115

4.2 Actions of quaternions, Lie groups and Lie algebras 116

4.2.1 AD, Ad, ad, Ad* and ad* actions of quaternions 117

4.2.2 AD, Ad, and ad for Lie algebras and groups 118

4.3 Example: The Heisenberg Lie group 124

4.3.1 Definitions for the Heisenberg group 124

4.3.2 Adjoint actions: AD, Ad and ad 126

4.3.3 Coadjoint actions: Ad* and ad* 127

4.3.4 Coadjoint motion and harmonic oscillations 129

5 The special orthogonal group SO(3) 131

5.1 Adjoint and coadjoint actions of SO(3) 132

5.1.1 Ad and ad operations for the hat map 132

5.1.2 AD, Ad and ad actions of SO(3) 133

5.1.3 Dual Lie algebra isomorphism 135

6 Adjoint and codajoint semidirect-product group actions 141

6.1 Special Euclidean group SE(3) 142

6.2 Adjoint operations for SE(3) 144

6.3 Adjoint actions of SE(3)'s Lie algebra 148

6.3.1 The ad action of se(3) on itself 148

6.3.2 The ad* action of se(3) on its dual se(3)* 149

6.3.3 Left versus right 151

6.4 Special Euclidean group SE(2) 153

6.5 Semidirect-product group SL(2,R) SR² 156

6.5.1 Definitions for SL(2,E)SR² 156

6.5.2 AD, Ad, and ad actions 158

6.5.3 Ad* and ad* actions 160

6.5.4 Coadjoint motion relation 162

6.6 Galilean group 164

6.6.1 Definitions for G(3) 164

6.6.2 AD, Ad, and ad actions of G(3) 165

6.7 Iterated semidirect products 167

7 Euler-Poincaré and Lie-Poisson equation SE(3) 169

7.1 Euler-Poincaré equations for left-invariant Lagrangians under SE(3) 170

7.1.1 Legendre transform from se(3) to se(3)* 172

7.1.2 Lie-Poisson bracket on se(3)* 172

7.1.3 Coadjoint motion on se(3)* 173

7.2 Kirchhoff equations on se(3)* 176

7.2.1 Looks can be deceiving: The heavy top 178

8 Heavy-top equation 181

8.1 Introduction and definitions 182

8.2 Heavy-top action principle 183

8.3 Lie-Poisson brackets 184

8.3.1 Lie-Poisson brackets and momentum maps 185

8.3.2 Lie-Poisson brackets for the heavy top 186

8.4 Clebsch action principle 187

8.5 Kaluza-Klein construction 188

9 The Euler-Poincaré theorem 193

9.1 Action principles on Lie algebras 194

9.2 Hamilton-Pontryagin principle 198

9.3 Clebsch approach to Euler-Poincaré 199

9.3.1 Defining the Lie derivative 201

9.3.2 Clebsch Euler-Poincaré principle 202

9.4 Lie-Poisson Hamiltonian formulation 206

9.4.1 Cotangent-lift momentum maps 207

10 Lie-Poisson Hamiltonian form of a continuum spin chain 209

10.1 Formulating continuum spin chain equations 210

10.2 Euler-Poincaré equations 212

10.3 Hamiltonian formulation 213

11 Momentum maps 221

11.1 The momentum map 222

11.2 Cotangent lift 224

11.3 Examples of momentum maps 226

11.3.1 The Poincaré sphere S² ∈ S³ 237

11.3.2 Overview 242

12 Roudn, rolling rigid bodies 245

12.1 Introduction 246

12.1.1 Holonomic versus nonholonomic 246

12.1.2 The Chaplygin ball 248

12.2 Nonholonomic Hamilton-Pontryagin variational principle 252

12.2.1 HP principle for the Chaplygin ball 256

12.2.2 Circular disk rocking in a vertical plane 265

12.2.3 Euler's rolling and spinning disk 268

12.3 Nonholonomic Euler-Poincaré reduction 275

12.3.1 Semidirect-product structure 276

12.3.1 Euler-Poincaré theorem 278

12.3.3 Constrained reduced Lagrangian 282

A Geometrical structure of classical mechanics 287

A 1 Manifold 288

A.2 Motion: Tangent vectors and flows 296

A.2.1 Vector fields, integral curves and flows 297

A.2.2 Differentials of functions: The cotangent bundle 299

A.3 Tangent and cotangent lifts 300

A.3.1 Summary of derivatives on manifolds 301

B Lie groups and Lie algebras 305

B.1 Matrix Lie groups 306

B.2 Defining matrix Lie algebras 310

B.3 Examples of matrix Lie groups 312

B.4 Lie group actions 314

B.4.1 Left and right translations on a Lie group 316

B 5 Tangent and cotangent lift actions 317

B.6 Jacobi-Lie bracket 320

B.7 Lie derivative and Jacobi-Lie bracket 323

B.7.1 Lie derivative of a vector field 325

B.7.2 Vector fields in ideal fluid dynamics323

C Enhanced coursework 327

C.1 Variations on rigid-body dynamics 328

C.1.1 Two times 328

C.1.2 Rotations in complex space 334

C.1.3 Rotations in four dimensions: SO(4) 338

C.2 C³ oscillators 343

C.3 Momentum maps for GL(n,R) 348

C.4 Motion on the symplectic Lie group Sp(2) 354

C.5 Two coupled rigid bodies 359

D Poincaré's 1901 paper 363

Bibliography 367

Index 385

Editorial Reviews

These volumes are a valuable addition to the literature on geometric mechanics. They are very well written, can be used for various courses independently or in conjunction with other texts, both more elementary and advanced, and can also serve as a good starting point for someone interested in the subject and willing to learn the principles and techniques of geometric mechanics. -- MathSciNet "MathSciNet"

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