Geometry Of Nonholonomically Constrained Systems

by Richard H Cushman, Hans Duistermaat, Jedrzej Sniatycki

Overview

This book gives a modern differential geometric treatment of linearly nonholonomically constrained systems. It discusses in detail what is meant by symmetry of such a system and gives a general theory of how to reduce such a symmetry using the concept of a differential space and the almost Poisson bracket structure of its algebra of smooth functions. The above theory is applied to the concrete example of Carathéodory's sleigh and the convex rolling rigid body. The qualitative behavior of the motion of the rolling disk is treated exhaustively and in detail. In particular, it classifies all motions of the disk, including those where the disk falls flat and those where it nearly falls flat.The geometric techniques described in this book for symmetry reduction have not appeared in any book before. Nor has the detailed description of the motion of the rolling disk. In this respect, the authors are trail-blazers in their respective fields.

Product Details

ISBN-13:	9789814289481
Publisher:	World Scientific Publishing Company, Incorporated
Publication date:	12/11/2009
Series:	Advanced Series In Nonlinear Dynamics , #26
Pages:	424
Product dimensions:	6.10(w) x 9.00(h) x 0.90(d)

Table of Contents

Acknowledgments vii

Foreword ix

1 Nonholonomically constrained motions 1

1.1 Newton's equations 1

1.2 Constraints 3

1.3 Lagrange-d' Alembert equations 5

1.4 Lagrange derivative 7

1.5 Hamilton-d'Alembert equations 9

1.6 Distributional Hamiltonian formulation 13

1.6.1 The symplectic distribution (H, ϖ) 13

1.6.2 H and ϖ 16

1.6.3 Distributional Hamiltonian vector field 20

1.7 Almost Poisson brackets 24

1.7.1 Hamilton's equations 24

1.7.2 Nonholonomic Dirac brackets 28

1.8 Momenta and momentum equation 33

1.8.1 Momentum functions 33

1.8.2 Momentum equations 35

1.8.3 Homogeneous functions 37

1.8.4 Momenta as coordinates 38

1.9 Projection principle 39

1.10 Accessible sets 41

1.11 Constants of motion 43

1.12 Notes 46

2 Group actions and orbit spaces 49

2.1 Group actions 49

2.2 Orbit spaces 50

2.3 Isotropy and orbit types 52

2.3.1 Isotropy types 52

2.3.2 Orbit types 53

2.3.3 When the action is proper 54

2.3.4 Stratification by orbit types 55

2.4 Smooth structure on an orbit space 56

2.4.1 Differential structure 56

2.4.2 The orbit space as a differential space 59

2.5 Subcartesian spaces 62

2.6 Stratification of the orbit space by orbit types 64

2.6.1 Orbit types in an orbit space 64

2.6.2 Stratification of an orbit space 66

2.6.3 Minimality of S 67

2.7 Derivations and vector fields on a differential space 68

2.8 Vector fields on a stratified differential space 73

2.9 Vector fields on an orbit space 74

2.10 Tangent objects to an orbit space 76

2.10.1 Stratified tangent bundle 76

2.10.2 Zariski tangent bundle 77

2.10.3 Tangent cone 77

2.10.4 Tangent wedge 78

2.11 Notes 79

3 Symmetry and reduction 81

3.1 Dynamical systems with symmetry 81

3.1.1 Invariant vector fields 81

3.1.2 Reduction of symmetry 82

3.1.3 Reduction for a free and proper G-action 82

3.1.4 Reduction nonfree proper action 82

3.2 Nonholonomic singular reduction 84

3.3 Nonholonomic regular reduction 93

3.4 Chaplygin systems 97

3.5 Orbit types and reduction 102

3.6 Conservation laws 105

3.6.1 Momentum map 105

3.6.2 Gauge momenta 112

3.7 Lifted actions and the momentum equation 113

3.7.1 Lifted actions 113

3.7.2 Momentum equation 117

3.8 Notes 120

4 Reconstruction, relative equilibria and relative periodic orbits 123

4.1 Reconstruction 123

4.1.1 Reconstruction for proper free actions 123

4.1.2 Reconstruction for nonfree proper actions 125

4.1.3 Application to nonholonomic systems 125

4.2 Relative equilibria 126

4.2.1 Basic properties 126

4.2.2 Quasiperiodic relative equilibria 129

4.2.3 Runaway relative equilibria 132

4.2.4 Relative equilibria for nonfree actions 133

4.2.5 Other relative equilibria 134

4.2.6 Famlies of quasiperiodic relative equilibria 139

4.3 Relative periodic orbits 152

4.3.1 Basic properties 152

4.3.2 Quasiperiodic relative periodic orbits 153

4.3.3 Runaway relative periodic orbits 157

4.3.4 G-action is nonfree 157

4.3.5 Other relative periodic orbits 158

4.3.6 Families of quasiperiodic relative periodic orbits 160

4.4 Notes 171

5 Carathéodory's sleigh 173

5.1 Basic set up 173

5.1.1 Configuration space 173

5.1.2 Kinetic energy 174

5.1.3 Nonholonomic constraint 175

5.2 Equations of motion 176

5.2.1 Lagrange-d'Alembert equations 176

5.2.2 Nonholonomic Dirac brackets 177

5.2.3 Lagrange-d'Alembert in a trivialization 179

5.2.4 Almost Poisson bracket form 181

5.2.5 Distributional Hamiltonian system 183

5.3 Reduction of the E(2) symmetry 187

5.3.1 The E(2) symmetry 188

5.3.2 The momentum equation 189

5.3.3 E(2)-reduced equations of motion 192

5.4 Motion on the E(2) reduced phase space 196

5.5 Reconstruction 198

5.5.1 Relative equilibria 198

5.5.2 General motions 199

5.5.3 Motion of a material point on the sleigh 200

5.6 Notes 203

6 Convex rolling rigid body 205

6.1 Basic set up 205

6.2 Unconstrained motion 208

6.3 Constraint distribution 210

6.4 Constrained equations of motion 215

6.4.1 Vector field on D 215

6.4.2 Computation of H and ϖ in a trivialization 218

6.4.3 Distributional vector field in a trivialization 221

6.5 Reduction of the translational R² symmetry 222

6.5.1 The R²-reduced equations of motion 223

6.5.2 Comparison with the Euler-Lagrange equations 224

6.5.3 The R²-reduced distribution &Hbar;_&Dbar;N and the 2-form ϖ_&Dbar;N 226

6.6 Reduction of E(2) symmetry 230

6.6.1 E(2) symmetry 231

6.6.2 E(2)-orbit space 233

6.6.3 E(2)-reduced distribution and 2-form 235

6.6.4 Reduced distributional system 240

6.7 Body of revolution 243

6.7.1 Geometric and dynamic symmetry 244

6.7.2 Reduction of the induced axial symmetry 248

6.7.3 Axially reduced equations of motion 249

6.8 Notes 262

7 The rolling disk 265

7.1 General set up 267

7.2 Reduction of the E(2) ? S¹ symmetry 269

7.2.1 First E(2), then S¹ 270

7.2.2 First S¹, then E(2) 274

7.3 Reconstruction 276

7.3.1 The E(2)-reduced flow 276

7.3.2 The full motion 277

7.3.3 The S¹-reduced flow 278

7.3.4 Geometry of the E(2) ? S¹ reduction map 280

7.4 Relative equilibria 283

7.4.1 The manifold of relative equilibria 283

7.4.2 One parameter groups 284

7.4.3 Angular speeds in terms of invariants 285

7.4.4 Motion of the relative equilibria 287

7.4.5 Nearly flat relative equilibria 290

7.5 A potential function on an interval 290

7.5.1 Chaplygin's equations 290

7.5.2 A conservative Newtonian system 292

7.5.3 Qualitative behavior 293

7.5.4 A special case of falling flat 295

7.6 Scaling 296

7.7 Solutions of the rescaled Chaplygin equations 298

7.7.1 The recessive solution 298

7.7.2 Asymptotics 299

7.7.3 The normalized even and odd solutions 301

7.7.4 Computation of r(0) and r′(0) 301

7.8 Bifurcations of a vertical disk 303

7.8.1 Degenerate equilibria 303

7.8.2 Vertical degenerate relative equilibria 304

7.8.3 Normal form of the potential 306

7.8.4 Cusps of the degeneracy locus 310

7.9 The global geometry of the degeneracy locus 311

7.9.1 The circle of degenerate critical points 312

7.9.2 A global description of the degeneracy locus 315

7.10 Falling flat 319

7.10.1 When the disk does not fall flat 320

7.10.2 When the disk falls flat 322

7.10.3 Limiting behavior when falling flat 324

7.11 Near falling flat 326

7.11.1 Elastic reflection 326

7.11.2 The increase of the angles ψ and χ 328

7.11.3 Motions near falling flat 332

7.12 The bifurcation diagram 340

7.12.1 The bifurcation set β 340

7.12.2 Off the bifurcation set β 341

7.12.3 On a coordinate axis or in an open quadrant 343

7.12.4 Near &script;_± 346

7.12.5 Global qualitative description of V_{&sigmabar;3, &sigmabar;4} 347

7.12.6 Global description of the orbits X_{&sigmabar;3, &sigmabar;4} 349

7.13 The integral map 351

7.13.1 Regular values of I 352

7.13.2 The global geometry of the critical value surface 353

7.14 Constant energy slices 358

7.14.1 Numerical pictures of the constant energy slices 358

7.14.2 Geometric features of the constant energy slices 362

7.14.3 Outward radial growth 362

7.14.4 The swallow tail sections 365

7.14.5 Behavior of cusp points 365

7.14.6 Over the coordinate axes in the (±₃, ±₄)-plane 367

7.14.7 Σ over &script;_plus-minus; 368

7.15 The spatial rotational shift 370

7.15.1 The shift 370

7.15.2 Quasiperiodic motion 371

7.15.3 The spatial rotational shift 372

7.15.4 Near elliptic relative equilibria 376

7.15.5 Nearly flat solutions 381

7.16 Notes 384

Bibliography 387

Index 395