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Symmetry and Perturbation Theory in Nonlinear Dynamics : Lecture Notes in Physics Monographs - Giampaolo Cicogna

Symmetry and Perturbation Theory in Nonlinear Dynamics

Lecture Notes in Physics Monographs

Hardcover Published: November 1999
ISBN: 9783540659044
Number Of Pages: 212

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This book deals with the theory of Poincare--Birkhoff normal forms, studying symmetric systems in particular. Attention is focused on general Lie point symmetries, & not just on symmetries acting linearly. Some results on the simultaneous normalization of a vector field describing a dynamical system & vector fields describing its symmetry are presented & a perturbative approach is also used. Attention is given to the problem of convergence of the normalizing transformation in the presence of symmetry, with some other extensions of the theory. The results are discussed for the general case of dynamical systems & also for the specific Hamiltonian setting.

Introductionp. 1
Definitions and Propertiesp. 5
Notationsp. 5
The Restricted Problem: Equationsp. 5
Periodic Solutions and Periodic Orbitsp. 6
The Period-in-Familyp. 8
Structure of Familiesp. 9
Family Segmentsp. 12
Symmetryp. 12
Stabilityp. 13
Generating Orbits and Keplerian Orbitsp. 14
Speciesp. 16
Generating Orbits of the First Speciesp. 21
Kindsp. 21
First Kindp. 22
The Case n = 1p. 22
The Case n &neq; 1p. 23
Second Kindp. 24
Symmetric Orbitsp. 26
Asymmetric Orbitsp. 31
Summaryp. 33
Generating Orbits of the Second Speciesp. 35
Arcsp. 35
Supporting Ellipses and Typesp. 38
Type 1p. 41
S-Arcsp. 42
The (A, Z) Planep. 48
S-Arc Familiesp. 52
T-Arcs and T-Arc Familiesp. 55
Overviewp. 59
Types 2, 3, 4p. 59
Ends of S- and T-Arc Familiesp. 62
Extremums of C: Arc Family Segmentsp. 68
Orbitsp. 73
Second Species Familiesp. 75
Ends of Family Segmentsp. 76
Generating Orbits of the Third Speciesp. 79
¿-Generating Orbitsp. 79
Continuation from ¿ = 0 to ¿ > 0p. 82
First Speciesp. 82
Second Speciesp. 82
¿-Generating Orbits for 0 < ¿ < 1/3p. 83
First Speciesp. 84
Second Speciesp. 85
Third Speciesp. 88
¿-Generating Orbits for ¿ = 1/3p. 88
Continuation from ¿ < 1/3 to ¿ = 1/3p. 91
Family fp. 91
Family ap. 91
Family cp. 92
Family g'p. 92
Family gp. 92
Recapitulationp. 92
Other Familiesp. 93
¿-Generating Orbits for ¿ > 1/3p. 93
Continuation from ¿ = 1/3 to ¿ > 1/3p. 94
Conclusionsp. 94
Bifurcation Orbitsp. 95
First Species and First Speciesp. 96
First Kind and First Kindp. 96
First Kind and Second Kind, Symmetricp. 96
First Kind and Second Kind, Asymmetricp. 97
Second Kind, Symmetric and Symmetricp. 97
Second Kind, Symmetric and Asymmetricp. 97
Second Kind, Asymmetric and Asymmetricp. 97
Second Species and First or Second Speciesp. 98
Total Bifurcationp. 99
Partial Bifurcationp. 115
Third Species and First or Second Speciesp. 123
Third Species and Third Speciesp. 123
Recapitulationp. 123
Junctions: Symmetryp. 125
First Species Bifurcationsp. 125
Third Species Bifurcationsp. 126
Second Species Bifurcationsp. 126
Partial Bifurcationp. 127
Total Bifurcationp. 128
Conclusionsp. 134
Junctions: Broucke's Principlep. 137
Definitionp. 137
Side of Passage for a Nodep. 139
Type 1p. 140
Type 2p. 143
Type 3p. 145
Side of Passage for an Antinodep. 147
Type 1p. 147
Type 2p. 150
Type 3p. 151
Results: Partial Bifurcationp. 152
Type 1p. 152
Type 2p. 160
Type 3p. 162
Results: Total Bifurcationp. 162
Type 1p. 162
Type 2p. 167
Type 3p. 167
Recapitulationp. 167
Fragmentsp. 171
Introductionp. 171
Accidentsp. 172
Explanation of the Fragment Tablesp. 173
First Species Family Segmentsp. 174
First Kind: Retrograde Orbitsp. 174
First Kind: Direct Orbitsp. 174
Second Kindp. 175
Second Species Family Segmentsp. 179
Detection of Bifurcation Orbitsp. 179
Data on Arc Family Segmentsp. 183
Fragmentsp. 190
Third Species Family Segmentsp. 196
Generating Familiesp. 203
Algorithmp. 203
Explanation of the Tablesp. 205
Natural Familiesp. 206
Family ap. 207
Family bp. 208
Family cp. 209
Family fp. 209
Family gp. 210
Family hp. 211
Family ip. 213
Family lp. 214
Family mp. 214
Summaryp. 214
Other Familiesp. 214
Comparison with Computed Familiesp. 220
Family ap. 221
Family bp. 223
Family cp. 224
Family fp. 225
Family gp. 225
Family hp. 225
Family ip. 225
Families l and mp. 229
Final Commentsp. 232
Correspondence Between Old and New Notationsp. 235
Arcsp. 235
Arc Familiesp. 237
Critical Arcsp. 240
The Domain D2p. 243
The Curved Boundary ¿p. 243
S-Arc Families in Domain D2p. 245
Number of Branchesp. 251
Partial Bifurcationp. 251
Type 1p. 251
Type 2p. 253
Type 3p. 254
Total Bifurcationp. 255
Type 1p. 256
Type 2p. 258
Type 3p. 260
Conclusionsp. 264
Index of Definitionsp. 265
Index of Notationsp. 269
Referencesp. 275
Table of Contents provided by Publisher. All Rights Reserved.

ISBN: 9783540659044
ISBN-10: 3540659048
Series: Lecture Notes in Physics Monographs
Audience: General
Format: Hardcover
Language: English
Number Of Pages: 212
Published: November 1999
Publisher: SPRINGER VERLAG GMBH
Country of Publication: DE
Dimensions (cm): 23.39 x 15.6  x 1.42
Weight (kg): 0.5

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