The Kepler problem: group theoretical aspects, regularization and quantization, with application to the study of perturbations
"The accompanying CD-ROM contains mainly the Microsoft Windows program KEPLER which calculates the effects of any perturbation of the Kepler problem and plots the resulting trajectories." -- p. [4] of cover.
Saved in:
| Main Author: | |
|---|---|
| Format: | Book |
| Language: | English |
| Published: |
Basel [u.a.]
Birkhäuser
2003
|
| Series: | Progress in mathematical physics
29 |
| Subjects: | |
| Online Access: | Inhaltsverzeichnis |
| Summary: | "The accompanying CD-ROM contains mainly the Microsoft Windows program KEPLER which calculates the effects of any perturbation of the Kepler problem and plots the resulting trajectories." -- p. [4] of cover. |
| Item Description: | Includes bibliographical references and index |
| Physical Description: | XVII, 439 S. graph. Darst. 1 CD-ROM |
| ISBN: | 0817669027 3764369027 |
Staff View
MARC
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| 100 | 1 | |a Cordani, Bruno |e Verfasser |0 (DE-588)124183069 |4 aut | |
| 245 | 1 | 0 | |a The Kepler problem |b group theoretical aspects, regularization and quantization, with application to the study of perturbations |c Bruno Cordani |
| 264 | 1 | |a Basel [u.a.] |b Birkhäuser |c 2003 | |
| 300 | |a XVII, 439 S. |b graph. Darst. |e 1 CD-ROM | ||
| 336 | |b txt |2 rdacontent | ||
| 337 | |b n |2 rdamedia | ||
| 338 | |b nc |2 rdacarrier | ||
| 490 | 1 | |a Progress in mathematical physics |v 29 | |
| 500 | |a Includes bibliographical references and index | ||
| 520 | 3 | |a "The accompanying CD-ROM contains mainly the Microsoft Windows program KEPLER which calculates the effects of any perturbation of the Kepler problem and plots the resulting trajectories." -- p. [4] of cover. | |
| 650 | 7 | |a FÍSICA MATEMÁTICA |2 larpcal | |
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Record in the Search Index
| _version_ | 1804129679375859713 |
|---|---|
| adam_text | Contents
Preface
vii
list of Figures
xv
1
Introductory Survey
1
1.1
Parti
-
Elementary Theory
.................... 2
1.1.1
Basic Facts
......................... 2
1.1.2
Separation of Variables and Action-Angle Variables
3
1.1.3
Quantization of the Kepler Problem
.......... 4
1.1.4
Regularization and Symmetry
.............. 5
1.2
Part II
-
Group-Geometric Theory
............... 5
1.2.1
Conformai
Regularization
................ 5
1.2.2
Spinorial Regularization
................. 7
1.2.3
Return to Separation of Variables
........... 8
1.2.4
Geometric Quantization
................. 9
1.2.5
Kepler Problem with a Magnetic
Monopole...... 10
1.3
Part III
-
Perturbation Theory
.................. 11
1.3.1
General Perturbation Theory
.............. 11
1.3.2
Perturbations of the Kepler Problem
......... 12
1.3.3
Perturbations with Axial Symmetry
.......... 13
_________________Contents
1.4
Part IV
-
Appendices
....................... 14
1.4.1
Differential Geometry
................... 14
1.4.2
Lie Groups and Lie Algebras
............... 15
1.4.3
Lagrangian Dynamics
................... 15
1.4.4
Hamiltonian Dynamics
.................. 16
I Elementary Theory
17
Basic Facts
18
2.1
Conies
................................ 18
2.2
Properties of the Keplerian Motion
............... 22
2.2.1
Energy
Я< О
........................ 23
2.2.2
Energy
Я
> 0........................ 25
2.2.3
Energy
Я
= 0........................ 26
2.3
The Three Anomalies
....................... 27
2.3.1
Energy
Я
< 0........................ 27
2.3.2
Energy
Я
> 0........................ 27
2.3.3
Energy
Я
= 0........................ 28
2.4
The Elements of the Orbit for
Ж
0.............. 29
2.5
The Repulsive Potential
...................... 32
Appendix
2.A The Kepler Equation
........................ 34
Separation of Variables and Action-Angle Coordinates
36
3.1
Separation of Variables
...................... 37
3.1.1
Spherical Coordinates
.................. 37
3.1.2
Parabolic Coordinates
.................. 38
3.1.3
Elliptic Coordinates
.................... 40
3.1.4
Spheroconical Coordinates
............... 42
3.2
Action-Angle Variables
...................... 44
3.2.1
Delaunay and
Poincaré
Variables
............ 44
3.2.2 Pauli
Variables
....................... 55
3.2.3
Monodramy
......................... 59
Quantization of the Kepler Problem
61
4.1
The
Schrödinger
Quantization
................. 61
4.1.1
Spherical Coordinates
.................. 66
4.1.2
Parabolic Coordinates
.................. 71
4.1.3
Elliptic Coordinates
.................... 73
4.1.4
Spheroconical Coordinates
............... 75
4.2 Pauli
Quantization
......................... 78
4.2.1
Canonical Quantization
................. 78
Contents xi
4.2.2 Pauli
Quantization
..................... 80
4.3 Fock
Quantization.........................
81
Appendix
4.A
Mathematical
Review....................... 87
4.A.1
Second
Order Linear Differential
Equations
..... 87
4.
Α.
2 Laplacian
on the Sphere and Homogeneous
Harmonic Polynomials
.................. 89
4.A.3 Associated Legendre Functions
............. 92
4.A.4 Generalized Laguerre Polynomials
........... 93
4.A.5 Surface Measure on the Sphere and Gamma
Function
........................... 94
4.A.6 Green Function of the Laplacian
............ 95
5
Regularization and Symmetry
96
5.1 Moser
Method
............................ 97
5.2
Souriau Method
........................... 102
5.2.1
Fock Parameters
...................... 104
5.2.2 Bacry-Györgyi
Parameters
................ 105
5.3
Kustaanheimo-Stiefel Transformation
............ 105
II Group-Geometric Theory
109
6
Conformai
Regularization
110
6.1
The
Conformai
Group
.......................
Ill
6.2
The Compactified Minkowski Space
.............. 115
6.3
The Cotangent Bundle to Minkowski Space
.......... 119
6.4
Regularization of the Kepler Problem
............. 129
7
Spinorial Regularization
143
7.1
TheHomomorphismSU(2,2)
-
SO(2,4)
........... 143
7.1.1
Two Bases for su(2,
2).................. 145
7.1.2
SU(2,
2)
and Compactified Minkowski Space
.... 147
7.2
Return to the Kustaanheimo-Stiefel Map
........... 150
7.3
Generalized Kustaanheimo-Stiefel Map
............ 156
8
Return to Separation of Variables
161
8.1
Separable Orthogonal Systems
................. 161
8.1.1 Stäckel
Theorem
...................... 162
8.1.2 Eisenhart
Theorem
.................... 164
8.1.3
Robertson Theorem
.................... 169
8.2
Finding Coordinate Systems Separating Kepler Problem
. 170
8.2.1
Spherical Coordinates
.................. 173
xii
Contents
8.2.2
Parabolic Coordinates
.................. 173
8.2.3
Elliptic Coordinates
.................... 175
8.2.4
Spheroconical Coordinates
............... 176
8.3
Integrable
Perturbations
..................... 177
8.3.1
Euler
Problem
....................... 179
8.3.2
Stark Problem
....................... 189
Appendix
8.A Jacobian Elliptic Functions
.................... 190
9
Geometric Quantization
192
9.1
Multiplier Representations
.................... 194
9.2
Quantization of Geodesies on the Sphere
........... 197
9.3
Quantization of the Kepler Problem
.............. 205
10
Kepler Problem with Magnetic
Monopole
211
10.1
Nonnull Twistors and Magnetic
Monopoles
......... 212
10.1.1
Bound Motions
....................... 219
10.1.2
Unbound Motions
..................... 222
10.1.3
Quantization
........................ 223
10.2
The MICZ System
.......................... 225
10.3
The Taub-NUT System
...................... 228
10.4
The BPST
Instanton
........................ 232
III Perturbation Theory
235
11
General Perturbation Theory
236
11.1
Formal Expansions
......................... 237
11.1.1
Lie Series and Formal Canonical Transformations
. 237
11.1.2
Homological Equation and its Formal Solution
... 242
11.2
The Convergence Problem
.................... 245
11.2.1
Convergence of Lie Series
................ 247
11.2.2
Homological Equation and its Solution
........ 249
11.2.3
Kolmogorov Theorem
................... 253
11.2.4
Nekhoroshev Theorem
.................. 262
Appendices
lLAResults from Diophantine Theory
............... 264
ll.BCauchy Inequality
......................... 265
12
Perturbations of the Kepler Problem
268
12.1
A More Convenient Hamiltonian
................ 270
12.2
Normalization (or Averaging) Method
............. 276
Contents xiii
12.3
Numerical
Integration....................... 284
12.3.1
Symbolic
Manipulation.................. 285
12.3.2
Compiling Equations
................... 288
Appendices
12.
A Variation of the Constants
.................... 291
12.BThe Stabilization Method
..................... 291
13
Perturbations with Axial Symmetry
293
13.1
Reduction of Orbit Manifold
................... 293
13.2
Lunar Problem
........................... 302
13.3
Stark and Quadratic
Zeeman
Effect
.............. 311
13.4
Satellite around Oblate Primary
................. 313
IV Appendices
321
A Differential Geometry
322
A.1 Rudiments of Topology
...................... 322
A.2 Differentiable Manifolds
..................... 324
A.2.1 Definition
.......................... 324
A.2.2 Tangent and Cotangent Spaces
............. 327
A.2.3 Push-forward and Pull-back
.............. 329
A.3 Tensors and Forms
........................ 331
A.3.1 Tensors
........................... 331
A.3.
2
Forms and Exterior Derivatives
............. 332
A.3.3 Lie Derivative
........................ 335
A.
3.4
Integration of Differential Forms
............ 337
A.4 Distributions and Frobenius Theorem
............. 341
A.5 Riemannian, Symplectic and
Poisson
Manifolds
....... 344
A.5.1 Riemannian Manifolds
.................. 344
A.5.
2
Symplectic Manifolds
................... 348
A.5.3
Poisson
Manifolds
..................... 352
A.6 Fibre Bundles
............................ 354
A.6.1 Definition
.......................... 354
A.6.
2
Principal and Associated Fibre Bundles
........ 357
В
lie Groups and lie Algebras
362
B.I Definition and Properties
..................... 362
B.2 Adjoint and Coadjoint Representation
............ 366
B.3 Action of a Lie Group on a Manifold
.............. 369
B.4 Classification of Lie Groups and Lie Algebras
........ 372
B.5 Connection on a Principal Bundle
............... 375
xiv_____________________________
Contents
С
Lagrangian Dynamics
378
C.I
Lagrange
Equations
........................ 378
C.2 Hamilton Principle
......................... 382
C.3 Noether Theorem
......................... 383
C.4 Reduced Lagrangian and Maupertuis Principle
....... 384
D Hamiltonian
Dynamics
388
D.I From
Lagrange
to Hamilton
................... 388
D.2 The Hamilton-Jacobi Integration Method
........... 390
D.2.1 Canonical Transformations
............... 390
D.2.
2
Hamilton-Jacobi Equation
................ 392
D.2.3 Geometric Description
.................. 393
D.2.4 The Time-dependent Case
................ 397
D.3 Symmetries and Reduction
................... 398
D.3.1 The Moment Map
..................... 399
D.3.2 Reduction of Symplectic Manifolds
.......... 402
D.3.3 Reduction of
Poisson
Manifolds
............ 404
D.4 Action-Angle Variables
...................... 408
D.4.1 Arnold Theorem
...................... 409
D.4.2 Degenerate Systems
.................... 416
D.4.3 Monodromy
......................... 417
Bibliography
423
Index
433
|
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| id | DE-604.BV015022613 |
| illustrated | Illustrated |
| indexdate | 2024-07-09T19:08:58Z |
| institution | BVB |
| isbn | 0817669027 3764369027 |
| language | English |
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| publishDate | 2003 |
| publishDateSearch | 2003 |
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| publisher | Birkhäuser |
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| series | Progress in mathematical physics |
| series2 | Progress in mathematical physics |
| spelling | Cordani, Bruno Verfasser (DE-588)124183069 aut The Kepler problem group theoretical aspects, regularization and quantization, with application to the study of perturbations Bruno Cordani Basel [u.a.] Birkhäuser 2003 XVII, 439 S. graph. Darst. 1 CD-ROM txt rdacontent n rdamedia nc rdacarrier Progress in mathematical physics 29 Includes bibliographical references and index "The accompanying CD-ROM contains mainly the Microsoft Windows program KEPLER which calculates the effects of any perturbation of the Kepler problem and plots the resulting trajectories." -- p. [4] of cover. FÍSICA MATEMÁTICA larpcal Kepler's laws Kepler-Bewegung (DE-588)4163577-2 gnd rswk-swf Kepler-Bewegung (DE-588)4163577-2 s DE-604 Progress in mathematical physics 29 (DE-604)BV013823265 29 Digitalisierung UB Regensburg application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=010091526&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | Cordani, Bruno The Kepler problem group theoretical aspects, regularization and quantization, with application to the study of perturbations Progress in mathematical physics FÍSICA MATEMÁTICA larpcal Kepler's laws Kepler-Bewegung (DE-588)4163577-2 gnd |
| subject_GND | (DE-588)4163577-2 |
| title | The Kepler problem group theoretical aspects, regularization and quantization, with application to the study of perturbations |
| title_auth | The Kepler problem group theoretical aspects, regularization and quantization, with application to the study of perturbations |
| title_exact_search | The Kepler problem group theoretical aspects, regularization and quantization, with application to the study of perturbations |
| title_full | The Kepler problem group theoretical aspects, regularization and quantization, with application to the study of perturbations Bruno Cordani |
| title_fullStr | The Kepler problem group theoretical aspects, regularization and quantization, with application to the study of perturbations Bruno Cordani |
| title_full_unstemmed | The Kepler problem group theoretical aspects, regularization and quantization, with application to the study of perturbations Bruno Cordani |
| title_short | The Kepler problem |
| title_sort | the kepler problem group theoretical aspects regularization and quantization with application to the study of perturbations |
| title_sub | group theoretical aspects, regularization and quantization, with application to the study of perturbations |
| topic | FÍSICA MATEMÁTICA larpcal Kepler's laws Kepler-Bewegung (DE-588)4163577-2 gnd |
| topic_facet | FÍSICA MATEMÁTICA Kepler's laws Kepler-Bewegung |
| url | http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=010091526&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
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