Global Aspects of Classical Integrable Systems

493 Pages·2015·6.166 MB·English

by  Richard H. Cushman, Larry M. Bates (auth.)

Richard H. Cushman
Larry M. Bates
Global Aspects
of Classical
Integrable
Systems
Second Edition

Richard H. Cushman Larry M. Bates
•
Global Aspects of Classical
Integrable Systems
Second Edition
Richard H.Cushman Larry M.Bates
Department ofMathematics andStatistics Department ofMathematics andStatistics
University of Calgary University of Calgary
Calgary Calgary
Canada Canada
1st edition1997byBirkhäuserVerlag, Switzerland
ISBN978-3-0348-0917-7 ISBN978-3-0348-0918-4 (eBook)
DOI 10.1007/978-3-0348-0918-4
LibraryofCongressControlNumber:2015939157
MathematicsSubjectClassification(2010):70-01,70E99,58F05,58-01
SpringerBaselHeidelbergNewYorkDordrechtLondon
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Table of Contents
Foreword
Introduction
Themathematicalpendulum .......................................xiii
Exercises ........................................................ xvi
Part I. Examples
I The harmonic oscillator
.
1.Hamilton’sequationsandS1symmetry ............................ 1
2.S1-energymomentummapping ................................... 3
3.U(2)-momentummapping ....................................... 7
4.TheHopffibration ............................................. 12
5.Invarianttheoryandreduction ................................... 20
6.Exercises ...................................................... 25
II Geodesics on S3
.
1.ThegeodesicandDelaunayvectorfields ......................... 31
2.TheSO(4)-momentummapping ................................. 34
3.TheKeplerproblem ............................................ 41
3.1TheKeplervectorfield .................................... 41
3.2Theso(4)-momentummap ................................ 43
3.3Kepler’sequation ......................................... 48
4RegularizationoftheKeplervectorfield .......................... 51
4.1Moser’sregularization ..................................... 51
4.2Ligon-Schaafregularization ................................ 56
5.Exercises ...................................................... 66
vi Tableofcontents
III The Euler top
.
1.FactsaboutSO(3) ............................................. 79
1.1Thestandardmodel ....................................... 79
1.2Theexponentialmap ...................................... 82
1.3Thesolidballmodel ...................................... 83
1.4Thespherebundlemodel .................................. 85
2.Leftinvariantgeodesics ......................................... 93
2.1Euler-Arnol’dequationsonSO(3)×R3 ..................... 93
2.2Euler-Arnol’dequationsonT S2×R3 ....................... 96
1
3.Symmetryandreduction ........................................ 99
3.1SO(3)symmetry .......................................... 99
3.2Constructionofthereducedphasespace ................... 101
3.3Geometryofthereductionmap ........................... 103
3.4Euler’sequations ........................................ 104
4.Qualitativebehaviorofthereducedsystem ...................... 106
5.Analysisoftheenergymomentummap ......................... 109
6.IntegrationoftheEuler-Arnol’dequations ....................... 114
7.Therotationnumber .......................................... 118
7.1Ananalyticformula ...................................... 118
7.2Poinsot’sconstruction .................................... 119
8.Atwistingphenomenon ....................................... 127
9.Exercises .................................................... 130
IV The spherical pendulum
.
1.Liouvilleintegrability ......................................... 139
2.ReductionoftheS1symmetry .................................. 142
2.1TheorbitspaceTR3/S1 .................................. 142
2.2Thesingularreducedspace ............................... 144
2.3DifferentialstructureonP ............................... 146
j
2.4PoissonbracketsonC∞(P) .............................. 148
j
2.5DynamicsonthereducedspaceP ........................ 150
j
3.Theenergymomentummapping ............................... 150
3.1CriticalpointsofEM ................................... 151
3.2CriticalvaluesofEM .................................... 153
3.3LevelsetsofthereducedHamiltonianH |P ................ 156
j j
3.4LevelsetsoftheenergymomentummappingEM .......... 157
4.Rotationnumberandfirstreturntime ........................... 167
4.1Definitionoffirstreturntimeandrotationnumber ........... 167
4.2Analyticpropertiesoftherotationnumber .................. 171
4.3Analyticpropertiesofthefirstreturntime .................. 175
5.Monodromy .................................................. 176
5.1Definitionofmonodromy ................................. 176
5.2Monodromyofthebundleofperiodlattices ................ 177
6.Exercises .................................................... 182
Tableofcontents vii
V The Lagrange top
.
1.Thebasicmodel .............................................. 193
2.Liouvilleintegrability ......................................... 194
3.ReductionoftherightS1-action ................................ 197
3.1ReductiontotheEuler-Poissonequations .................. 197
3.2Themagneticsphericalpendulum ......................... 202
4.ReductionoftheleftS1-action ................................. 205
4.1InducedactiononPa ..................................... 205
4.2Theorbitspace(Ja)−1(b)/S1 ............................. 207
(cid:2)
4.3Somedifferentialspaces .................................. 211
4.4PoissonstructureonC(cid:2)∞(Pa) .............................. 215
b
5.TheEuler-Poissonequations ................................... 224
5.1Thetwicereducedsystem ................................ 224
5.2Theenergymomentummapping .......................... 231
5.3Motionofthetipofthefigureaxis ......................... 235
6.Theenergymomentummapping ............................... 248
6.1TopologyofEM−1(h,a,b)andH−1(h) ................... 248
6.2Thediscriminantlocus ................................... 255
6.3Theperiodlattice ........................................ 262
6.4Monodromy ............................................. 266
7.TheHamiltonianHopfbifurcation .............................. 268
7.1ThelinearHamiltonianHopfbifurcation ................... 268
7.2ThenonlinearHamiltonianHopfbifurcation ................ 273
8.Exercises .................................................... 278
Part II. Theory
VI Fundamental concepts
.
1.Symplecticlinearalgebra ...................................... 281
2.Symplecticmanifolds ......................................... 285
3.Hamilton’sequations .......................................... 290
4.Poissonalgebrasandmanifolds ................................ 294
5.Exercises .................................................... 304
viii Tableofcontents
VII Systems with symmetry
.
1.Smoothgroupactions ......................................... 305
2.Orbitspaces .................................................. 310
2.1Orbitspaceofaproperaction ............................. 310
2.2Orbitspaceofaproperfreeaction ......................... 312
3.Differentialspaces ............................................ 318
3.1Differentialstructure ..................................... 318
3.2Anorbitspaceasadifferentialspace ...................... 321
3.3Subcartesianspaces ...................................... 323
3.4Stratificationofanorbitspacebyorbittypes ............... 327
3.5MinimalityofS ........................................ 329
4.Vectorfieldsonadifferentialspace ............................. 330
4.1Definitionofavectorfield ................................ 330
4.2Vectorfieldonastratifieddifferentialspace ................ 332
4.3Vectorfieldsonanorbitspace ............................ 333
5.Momentummappings ......................................... 335
5.1Generalproperties ....................................... 335
5.2Normalform ............................................ 341
6.Regularreduction ............................................. 346
6.1Thestandardapproach ................................... 346
6.2Analternativeapproach .................................. 349
7.Singularreduction ............................................ 351
7.1Singularreducedspaceanddynamics ...................... 351
7.2Stratificationofthesingularreducedspace ................. 359
8.Exercises .................................................... 366
VIII Ehresmann connections
.
1.Basicproperties .............................................. 373
2.TheEhresmanntheorems ...................................... 378
3.Exercises .................................................... 380
IX Action angle coordinates
.
1.Liouvilleintegrablesystems ................................... 385
2.Localactionanglecoordinates ................................. 385
3.Exercises .................................................... 390
X Monodromy
.
1.Theperiodlatticebundle ...................................... 391
2.Thegeometricmondromytheorem ............................. 393
2.1Thesingularfiber ........................................ 394
2.2Nearbysingularfibers .................................... 398
Tableofcontents ix
2.3Monodromy ............................................. 399
3.Thehyperboliccircularbilliard ................................. 402
3.1Thebasicmodel ......................................... 402
3.2ReductionoftheS1symmetry ............................ 404
3.3Partialreconstruction .................................... 405
3.4Fullreconstruction ....................................... 406
3.5Thefirstreturntimeandrotationangle ..................... 407
3.6Theactionfunctions ..................................... 409
4.Exercises .................................................... 410
XI Basic Morse theory
.
1.Preliminaries ................................................. 423
2.TheMorselemma ............................................ 426
3.TheMorseisotopylemma ..................................... 429
4.Exercise ..................................................... 430
Notes
ForwordandIntroduction ........................................ 431
Harmonicoscillator ............................................. 431
GeodesicsonS3 ................................................ 432
Eulertop ....................................................... 433
Sphericalpendulum ............................................. 435
Lagrangetop ................................................... 436
Fundamentalconcepts ........................................... 438
Systemswithsymmetry .......................................... 439
Ehresmannconnections .......................................... 441
Actionanglecoordinates ......................................... 442
Monodromy .................................................... 442
Morsetheory ................................................... 444
References
........................................................ 445
Acknowledgments
.............................................. 464
Index
............................................................... 465
Foreword
This book gives a complete global geometric description of the motion of the two di-
mensionalharmonicoscillator,theKeplerproblem,theEulertop,thesphericalpendulum
andtheLagrangetop.TheseclassicalintegrableHamiltoniansystemsoneseestreatedin
almosteveryphysicsbookonclassicalmechanics. Sowhyisthisbooknecessary? The
answeristhatthestandardtreatmentsarenotcomplete.Forinstanceinphysicsbooksone
cannotseethemonodromyinthesphericalpendulumfromitsexplicitsolutioninterms
ofellipticfunctionsnorcanonereadofffromtheexplicitsolutionthefactthatatennis
racketmakesanearhalftwistwhenitistossedsoastospinnearlyaboutitsintermediate
axis.Modernmathematicsbooksonmechanicsdonotusethesymplecticgeometrictools
theydeveloptotreatthequalitativefeaturesoftheseproblemseither. Onereasonforthis
is that their basic tool for removing symmetries of Hamiltonian systems, called regular
reduction,isnotgeneralenoughtohandleremovalofthesymmetrieswhichoccurinthe
sphericalpendulumorintheLagrangetop. Forthesesymmetriesoneneedssingularre-
duction. Anotherreasonisthattheobstructionstomakinglocalactionanglecoordinates
globalsuchasmonodromywerenotknownwhentheseworkswerewritten.
The point of view adopted in this book is to start with a somewhat unfamiliar abstract
mathematicalmodelofthephysicalsystemsuchasthestudyofthegeodesicflowofaleft
invariantmetriconthethreedimensionalrotationgroup.Usingthesymplecticgeometric
formulationofHamiltonianmechanicswethenshowthattheequationsofmotionagree
withthosefoundbymoretraditionalmethodsforawellknownphysicalsystem,namely,
theforcefreerigidbodyorEulertop. Thisjustifiesourmathematicalmodel. Wedonot
trytobuildourmodelfromfundamentalphysicalprinciples. Wehavenotwrittenabook
onmechanicsorHamiltonianparticledynamics. Weonlydiscussfivespecialintegrable
systems,whichisaverysmallsampleoftherichvarietyofgeneralHamiltoniansystems.
Moreoverthebehaviorofthesolutionsoftheseintegrablesystemsismuchmoreregular
thanthenearlyunpredictablemotionofageneralHamiltoniansystemsuchasthethree
bodyproblem.
Our main goal is to understand the global geometric features of our model integrable
systems. The main tool we use is reduction to remove the symmetries and to obtain
a system with one degree of freedom. This allows us to determine the range and the
topologyofeveryfiberoftheenergymomentummappingofthesystem. Thetopology
ofafibercorrespondingtoasingularvalueoftheenergymomentummappingisofgreat
interest. Physically,thesemotionsaresimplerthanthegeneralmotionandthereforeare
easiertostudyexperimentally.Mathematically,thesefiberscontainarelativeequilibrium
ofthesystem,thatis,amotionwhichisalsoanorbitofthesymmetrygroup.Forinstance,
inthesphericalpendulumtherelativeequilibriaarecircularorbitsonthe2-spherewhich