The book provides an updated overview of the recent developments in the various different fields of nonlinear dynamics, covering both theory and applications. Special emphasis is given to algebraic and geometric integrability, solutions to the N-body problem of the "choreography" type, geometry and symmetry of dynamical systems, integrable evolution equations, various different perturbation theories, and bifurcation analysis.
Author(s): Giuseppe Gaeta, Barbara Prinari, Stefan Rauch-Wojciechowski, Susanna Terracini
Publisher: World Scientific Publishing Company
Year: 2005
Language: English
Pages: 347
Papers......Page 10
Foreword......Page 6
References......Page 7
Acknowledgements......Page 8
1. Introduction......Page 14
2. Lie group actions and invariant vector fields......Page 15
3. Lie algebra cohomology......Page 16
4. A map on the G invariant de Rham complex......Page 17
5. The cochain condition......Page 19
6. Examples......Page 23
References......Page 25
1. Introduction......Page 26
2. Trihamiltonian framework......Page 27
3. Trihamiltonian extension of Toda lattice......Page 28
4. Trihamiltonian extension of natural separable systems......Page 31
References......Page 33
1. Introduction......Page 35
2. Two-Dimensional Wave Packets......Page 36
References......Page 39
1. Introduction......Page 40
2. The periodic solutions......Page 42
3.1. Numerical data generated by CONTENT......Page 43
3.2. Averaging method results......Page 44
4.1. The averaged system......Page 45
5 . Conclusions......Page 46
References......Page 47
1. Introduction......Page 48
2. Setting and notations......Page 49
3. Symmetries and coercivity......Page 50
4. Symmetry constraints for collision trajectories......Page 51
5. Collisions with the Isosceles Symmetry......Page 52
References......Page 55
1. Posing a question......Page 56
2. Equivalent systems......Page 57
3. The equivalence theorem of Levi-Civita......Page 58
4. Main theorems......Page 59
5 . L-tensors, L-sequences, L-systems......Page 61
6. Cofactor and bi-cofactor systems are L-systems......Page 62
References......Page 63
1. Introduction......Page 64
2. Shadowing infinite collision chains......Page 66
3. Twist map reformulation......Page 68
4. Almost autonomous case......Page 69
References......Page 71
1. Introduction......Page 72
2. Lie symmetries......Page 73
3. New exact solutions......Page 77
Acknowledgments......Page 78
References......Page 79
1. Introduction......Page 80
2. Singularly Perturbed Reversible Vector Fields on the plane......Page 82
3. Fold, Transcritical and Pitchfork Singularities......Page 84
References......Page 87
1. The vector of inertia......Page 88
3. Central and complex central configurations......Page 90
4. Some homographic solutions......Page 91
5. The three-body complex central configurations......Page 92
6. A particular case of the three-body problem with M = 0......Page 94
References......Page 95
1. Conformal Killing tensors and separation of variables......Page 96
2. Conformal Killing tensors and L-systems......Page 98
2.1. K-sequences and C-sequences......Page 99
2.2. C-sequences and L-systems......Page 101
References......Page 103
1. The functional class S......Page 104
2.1. Highly competing diffusion systems......Page 105
2.2. An optimal partition problem......Page 106
3. The regularity theory for the elements of S......Page 107
4.1. Local properties of the free boundary......Page 109
References......Page 111
Introduction......Page 112
1. Standard prolongations......Page 113
2.2. The work of Pucci and Saccomandi......Page 114
3. Mu-prolongations; mu-symmetries for PDEs......Page 115
4. Mu-symmetries for systems of PDEs......Page 116
5. Compatibility condition and gauge equivalence......Page 117
References......Page 118
1. Introduction......Page 119
2. Potential symmetries......Page 122
3. Linearization......Page 125
References......Page 127
1.1. The model......Page 128
1.2. The results......Page 129
2.1. Lyapunov-Schmidt decomposition......Page 130
2.2. Zero-order solution......Page 131
2.3. Avoiding the loss of regularity......Page 132
3.2. Counterterms and resummations......Page 133
References......Page 135
16. Fundamental covariants in the invariant theory of Killing tensors J . T. Homood, R. G. McLenaghan, R. G. Smirnov and D. The......Page 137
2. Invariant theory of Killing tensors (ITKT)......Page 138
3. The main result......Page 140
References......Page 144
17. Global geometry of 3-body trajectories with vanishing angular momentum W. Y. Hsiang......Page 145
1. Jacobi’s reformulation of Lagrange’s least action principle, a profound geometrization of mechanics......Page 146
2. Conservation of angular momentum and the orbital geometry of ( S 0 ( 3 ) , M )......Page 147
3. Kinematic geometry of rn-triangles......Page 148
4. Shape curves, cone surfaces and the geodesic equation of (M(U,h) ds2h)......Page 150
5. Formula of VU* and a monotonicity theorem......Page 151
6. Some problems on global geometry of geodesics in (M(U,h) da2h)......Page 152
References......Page 153
1. Preliminaries and examples......Page 154
1.1. Proposition......Page 155
1.3. Example:......Page 156
2. The study of the boundary of the set of the controllable affine systems C ,......Page 157
3. Study of border cases......Page 158
4. Main result......Page 161
References......Page 162
Appendix......Page 163
1. Equations of motion......Page 164
2. Action-angle variables......Page 166
3. Existence of conditionally-periodic motions......Page 168
References......Page 171
1. Introduction......Page 172
2. Expanding in q2......Page 177
3. Expanding in y......Page 180
4. Expanding in......Page 182
5 . Conclusions......Page 184
References......Page 186
1. Introduction and equations of motion......Page 188
2. The main result......Page 191
References......Page 195
1. Introduction......Page 196
2. General rational Lax operators......Page 197
3. The problem of reduction and the reduction group......Page 198
3.1. Reduction group and automorphic subalgebras......Page 199
4. DN-Reductions of sl(2, C)-Lax operators with simple poles......Page 200
4.1. Automorphic Lie algebras corresponding t o twisted (A-dependent) automorphisms......Page 202
References......Page 204
23. Geometric reduction of Poisson operators.. K. Marciniak and M. Btaszak......Page 206
References......Page 210
1.2. History......Page 211
2. Resent results......Page 213
3.1. Integrability fog. the geodesic flows of geodesically equivalent metrics......Page 214
3.2. What is special in these integrals?......Page 215
3.3. If the metrics are strictly non-proportional......Page 216
3.5. General case......Page 217
References......Page 219
2. The method......Page 222
3.1. Fold-Flap......Page 224
3.2. An example......Page 225
References......Page 227
1. Introduction......Page 228
2. Alignment......Page 229
3. Alignment polynomials......Page 231
4. Bivectors......Page 232
5 . Weyl-type tensors......Page 233
References......Page 235
1. Introduction......Page 236
2. RGS method to initial value problem......Page 237
3. Perfect gas dynamics......Page 238
References......Page 241
28. Refined computation of hypernormal forms.. J. Murdoclc......Page 242
References......Page 248
1. Introduction......Page 249
2. The concept of variational C" -symmetry......Page 250
3. Order reduction through variational Cm -symmetries......Page 251
4. Conservation of variational symmetries by order reduct ions......Page 253
5. Conclusions......Page 255
References......Page 256
1. Introduction......Page 257
2. Tameness of Lie Pseudogroups......Page 259
3. Regularity of Orbits......Page 263
References......Page 266
1. Introduction......Page 268
2. Natural packets: Two time scales......Page 270
3. Equipartition times......Page 271
3.2. Exponentially long times t o equipartition......Page 272
3.3. Thermodynamic limit......Page 273
Appendix: The model......Page 274
References......Page 275
1. Introduction......Page 276
2. Boussinesq equations modeling FPU chains......Page 277
3. Hamiltonian structure of the gB equation......Page 278
4. Averaging......Page 279
5 . Dimensional analysis......Page 280
References......Page 282
2. Lagrangian approach......Page 284
3. Hamiltonian approach......Page 287
4. Strong integrability......Page 288
5.1. Polar coordinates......Page 289
5.3. Elliptical coordinates......Page 290
References......Page 291
1. Introduction......Page 292
2. Classification of singularities......Page 293
3. The Orbit Space approach......Page 295
4.1. Bifurcation diagmms and classification of orbit spaces......Page 297
4.2. Boundary singularities......Page 300
5. Conclusions and outlooks......Page 302
References......Page 303
1. Introduction......Page 304
2. Lennard-Jones potential and symmetries......Page 305
2.1. Variational methods o n symmet~% paths......Page 306
3.1. Variational characterisation......Page 307
4. Simple non-trivial rotationally symmetric orbits......Page 308
5. Choreographies in Fourier space......Page 309
6. Conclusions......Page 311
References......Page 312
1. Introduction......Page 313
2. Variational approach......Page 314
3. Filamentation......Page 316
4. Beam carrying phase singularity......Page 317
References......Page 319
1. Introduction......Page 321
2. Differential Invariants for Infinite-Dimensional Poincarb-Type Algebra......Page 323
3. Conclusion......Page 324
References......Page 325
Conference information......Page 13
Conference program......Page 326
List of participants......Page 331
List of communications......Page 334
Workshop SPT96 - Torino, 15-20 December 1996 [1]......Page 339
Regular papers......Page 340
Tutorial papers......Page 341
Regular papers......Page 342
Conference SPT2002 - Cala Gonone, 19-26 May 2002 [5]......Page 343
References......Page 345
