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.
Author(s): Richard H. Cushman, Jedrzej Sniatycki
Year: 2009
Language: English
Pages: 421
Cover......Page 1
Title......Page 4
Copyright......Page 5
Dedication......Page 6
Acknowledgments ......Page 8
Foreword ......Page 10
Contents ......Page 12
1.1 Newton?ˉs equations ......Page 18
1.2 Constraints ......Page 20
1.3 Lagrange-d?ˉAlembert equations ......Page 22
1.4 Lagrange derivative ......Page 24
1.5 Hamilton-d?ˉAlembert equations ......Page 26
1.6.1 The symplectic distribution (H,$) ......Page 30
1.6.2 H and $ ......Page 33
1.6.3 Distributional Hamiltonian vector field ......Page 37
1.7.1 Hamilton?ˉs equations ......Page 41
1.7.2 Nonholonomic Dirac brackets ......Page 45
1.8.1 Momentum functions ......Page 50
1.8.2 Momentum equations ......Page 52
1.8.3 Homogeneous functions ......Page 54
1.8.4 Momenta as coordinates ......Page 55
1.9 Projection principle ......Page 56
1.10 Accessible sets ......Page 58
1.11 Constants of motion ......Page 60
1.12 Notes ......Page 63
2.1 Group actions ......Page 66
2.2 Orbit spaces ......Page 67
2.3.1 Isotropy types ......Page 69
2.3.2 Orbit types ......Page 70
2.3.3 When the action is proper ......Page 71
2.3.4 Stratification by orbit types ......Page 72
2.4.1 Differential structure ......Page 73
2.4.2 The orbit space as a differential space ......Page 76
2.5 Subcartesian spaces ......Page 79
2.6.1 Orbit types in an orbit space ......Page 81
2.6.2 Stratification of an orbit space ......Page 83
2.6.3 Minimality of S ......Page 84
2.7 Derivations and vector fields on a differential space ......Page 85
2.8 Vector fields on a stratified differential space ......Page 90
2.9 Vector fields on an orbit space ......Page 91
2.10.1 Stratified tangent bundle ......Page 93
2.10.3 Tangent cone ......Page 94
2.10.4 Tangent wedge ......Page 95
2.11 Notes ......Page 96
3.1.1 Invariant vector fields ......Page 98
3.1.4 Reduction nonfree proper action ......Page 99
3.2 Nonholonomic singular reduction ......Page 101
3.3 Nonholonomic regular reduction ......Page 110
3.4 Chaplygin systems ......Page 114
3.5 Orbit types and reduction ......Page 119
3.6.1 Momentum map ......Page 122
3.6.2 Gauge momenta ......Page 129
3.7.1 Lifted actions ......Page 130
3.7.2 Momentum equation ......Page 134
3.8 Notes ......Page 137
4.1.1 Reconstruction for proper free actions ......Page 140
4.1.3 Application to nonholonomic systems ......Page 142
4.2.1 Basic properties ......Page 143
4.2.2 Quasiperiodic relative equilibria ......Page 146
4.2.3 Runaway relative equilibria ......Page 149
4.2.4 Relative equilibria for nonfree actions ......Page 150
4.2.5 Other relative equilibria ......Page 151
4.2.6 Famlies of quasiperiodic relative equilibria ......Page 156
4.3.1 Basic properties ......Page 169
4.3.2 Quasiperiodic relative periodic orbits ......Page 170
4.3.4 G-action is nonfree ......Page 174
4.3.5 Other relative periodic orbits ......Page 175
4.3.6 Families of quasiperiodic relative periodic orbits ......Page 177
4.4 Notes ......Page 188
5.1.1 Configuration space ......Page 190
5.1.2 Kinetic energy ......Page 191
5.1.3 Nonholonomic constraint ......Page 192
5.2.1 Lagrange-d?ˉAlembert equations ......Page 193
5.2.2 Nonholonomic Dirac brackets ......Page 194
5.2.3 Lagrange-d?ˉAlembert in a trivialization ......Page 196
5.2.4 Almost Poisson bracket form ......Page 198
5.2.5 Distributional Hamiltonian system ......Page 200
5.3 Reduction of the E(2) symmetry ......Page 204
5.3.1 The E(2) symmetry ......Page 205
5.3.2 The momentum equation ......Page 206
5.3.3 E(2)-reduced equations of motion ......Page 209
5.4 Motion on the E(2) reduced phase space ......Page 213
5.5.1 Relative equilibria ......Page 215
5.5.2 General motions ......Page 216
5.5.3 Motion of a material point on the sleigh ......Page 217
5.6 Notes ......Page 220
6.1 Basic set up ......Page 222
6.2 Unconstrained motion ......Page 225
6.3 Constraint distribution ......Page 227
6.4.1 Vector field on D ......Page 232
6.4.2 Computation of H and $ in a trivialization ......Page 235
6.4.3 Distributional vector field in a trivialization ......Page 238
6.5 Reduction of the translational R2 symmetry ......Page 239
6.5.1 The R2-reduced equations of motion ......Page 240
6.5.2 Comparison with the Euler-Lagrange equations ......Page 241
6.5.3 The R2-reduced distribution HDN and the 2-form $DN ......Page 243
6.6 Reduction of E(2) symmetry ......Page 247
6.6.1 E(2) symmetry ......Page 248
6.6.2 E(2)-orbit space ......Page 250
6.6.3 E(2)-reduced distribution and 2-form ......Page 252
6.6.4 Reduced distributional system ......Page 257
6.7 Body of revolution ......Page 260
6.7.1 Geometric and dynamic symmetry ......Page 261
6.7.2 Reduction of the induced axial symmetry ......Page 265
6.7.3 Axially reduced equations of motion ......Page 266
6.8 Notes ......Page 279
7. The rolling disk ......Page 282
7.1 General set up ......Page 284
7.2 Reduction of the E(2) ?á S1 symmetry ......Page 286
7.2.1 First E(2), then S1 ......Page 287
7.2.2 First S1, then E(2) ......Page 291
7.3.1 The E(2)-reduced flow ......Page 293
7.3.2 The full motion ......Page 294
7.3.3 The S1-reduced flow ......Page 295
7.3.4 Geometry of the E(2) ?á S1 reduction map ......Page 297
7.4.1 The manifold of relative equilibria ......Page 300
7.4.2 One parameter groups ......Page 301
7.4.3 Angular speeds in terms of invariants ......Page 302
7.4.4 Motion of the relative equilibria ......Page 304
7.5.1 Chaplygin?ˉs equations ......Page 307
7.5.2 A conservative Newtonian system ......Page 309
7.5.3 Qualitative behavior ......Page 310
7.5.4 A special case of falling flat ......Page 312
7.6 Scaling ......Page 313
7.7.1 The recessive solution ......Page 315
7.7.2 Asymptotics ......Page 316
7.7.4 Computation of r(0) and r0(0) ......Page 318
7.8.1 Degenerate equilibria ......Page 320
7.8.2 Vertical degenerate relative equilibria ......Page 321
7.8.3 Normal form of the potential ......Page 323
7.8.4 Cusps of the degeneracy locus ......Page 327
7.9 The global geometry of the degeneracy locus ......Page 328
7.9.1 The circle of degenerate critical points ......Page 329
7.9.2 A global description of the degeneracy locus ......Page 332
7.10 Falling flat ......Page 336
7.10.1 When the disk does not fall flat ......Page 337
7.10.2 When the disk falls flat ......Page 339
7.10.3 Limiting behavior when falling flat ......Page 341
7.11.1 Elastic reflection ......Page 343
7.11.2 The increase of the angles and ......Page 345
7.11.3 Motions near falling flat ......Page 349
7.12.1 The bifurcation set B ......Page 357
7.12.2 Off the bifurcation set B ......Page 358
7.12.3 On a coordinate axis or in an open quadrant ......Page 360
7.12.4 Near `?à ......Page 363
7.12.5 Global qualitative description of V3,4 ......Page 364
7.12.6 Global description of the orbits of X3,4 ......Page 366
7.13 The integral map ......Page 368
7.13.1 Regular values of I ......Page 369
7.13.2 The global geometry of the critical value surface ......Page 370
7.14.1 Numerical pictures of the constant energy slices ......Page 375
7.14.3 Outward radial growth ......Page 379
7.14.5 Behavior of cusp points ......Page 382
7.14.6 Over the coordinate axes in the (3, 4)-plane ......Page 384
7.14.7 over `?à ......Page 385
7.15.1 The shift ......Page 387
7.15.2 Quasiperiodic motion ......Page 388
7.15.3 The spatial rotational shift ......Page 389
7.15.4 Near elliptic relative equilibria ......Page 393
7.15.5 Nearly flat solutions ......Page 398
7.16 Notes ......Page 401
Bibliography ......Page 404
Index ......Page 412
