The Kepler Problem

This book contains a comprehensive treatment of the Kepler problem, i.e., the two body problem. It is divided into four parts. In the first part, written at an undergraduate student level, the arguments are presented in an elementary fashion, and the properties of the problem are demonstrated in a purely computational manner. In the second part a unifying point of view, original to the author, is presented which centers the exposition on the intrinsic group-geometrical aspects. This part requires more mathematical background, which the reader will find in the fourth part, in particular, the basic tools of differential geometry and analytical mechanics used in the book. The third part exploits some results of the second part to give a geometrical description of the perturbation theory of the Kepler problem.

Each of the four parts, which are to some extent independent, could itself form the basis for a one-semester course. The accompanying CD contains mainly the Microsoft Windows program KEPLER developed by the author. This program calculates the effects of any perturbation of the Kepler problem and plots the resulting trajectories.

Texte du rabat

Because of the correspondences existing among all levels of reality, truths pertaining to a lower level can be considered as symbols of truths at a higher level and can therefore be the "foundation" or support leading by analogy to a knowledge of the latter. This confers to every science a superior or "elevating" meaning, far deeper than its own original one. - R. GUENON, The Crisis of Modern World Having been interested in the Kepler Problem for a long time, I have al­ ways found it astonishing that no book has been written yet that would address all aspects of the problem. Besides hundreds of articles, at least three books (to my knowledge) have indeed been published al­ ready on the subject, namely Englefield (1972), Stiefel & Scheifele (1971) and Guillemin & Sternberg (1990). Each of these three books deals only with one or another aspect of the problem, though. For example, En­ glefield (1972) treats only the quantum aspects, and that in a local way. Similarly, Stiefel & Scheifele (1971) only considers the linearization of the equations of motion with application to the perturbations of celes­ tial mechanics. Finally, Guillemin & Sternberg (1990) is devoted to the group theoretical and geometrical structure.

Contenu

Preface.- List of Figures.- 1 Introductory Survey.- 1.1 Part I - Elementary Theory.- 1.1.1 Basic Facts.- 1.1.2 Separation of Variables and Action-Angle Variables.- 1.1.3 Quantization of the Kepler Problem.- 1.1.4 Regularization and Symmetry.- 1.2 Part II - Group-Geometric Theory.- 1.2.1 Conformal Regularization.- 1.2.2 Spinorial Regularization.- 1.2.3 Return to Separation of Variables.- 1.2.4 Geometric Quantization.- 1.2.5 Kepler Problem with a Magnetic Monopole.- 1.3 Part III - Perturbation Theory.- 1.3.1 General Perturbation Theory.- 1.3.2 Perturbations of the Kepler Problem.- 1.3.3 Perturbations with Axial Symmetry.- 1.4 Part IV - Appendices.- 1.4.1 Differential Geometry.- 1.4.2 Lie Groups and Lie Algebras.- 1.4.3 Lagrangian Dynamics.- 1.4.4 Hamiltonian Dynamics.- I Elementary Theory 17.- 2 Basic Facts.- 2.1 Conics.- 2.2 Properties of the Keplerian Motion.- 2.2.1 Energy H 0.- 2.2.3 Energy H = 0.- 2.3 The Three Anomalies.- 2.3.1 Energy H 0.- 2.3.3 Energy H = 0.- 2.4 The Elements of the Orbit for H < 0.- 2.5 The Repulsive Potential.- Append.- 2.A The Kepler Equation.- 3 Separation of Variables and Action-Angle Coordinates.- 3.1 Separation of Variables.- 3.1.1 Spherical Coordinates.- 3.1.2 Parabolic Coordinates.- 3.1.3 Elliptic Coordinates.- 3.1.4 Spheroconical Coordinates.- 3.2 Action-Angle Variables.- 3.2.1 Delaunay and Poincaré Variables.- 3.2.2 Pauli Variables.- 3.2.3 Monodromy.- 4 Quantization of the Kepler Problem.- 4.1 The Schrödinger Quantization.- 4.1.1 Spherical Coordinates.- 4.1.2 Parabolic Coordinates.- 4.1.3 Elliptic Coordinates.- 4.1.4 Spheroconical Coordinates.- 4.2 Pauli Quantization.- 4.2.1 Canonical Quantization.- 4.2.2 Pauli Quantization.- 4.3 Fock Quantization.- Append.- 4.A Mathematical Review.- 4.A.1 Second Order Linear Differential Equations.- 4.A.2 Laplacian on the Sphere and Homogeneous Harmonic Polynomials.- 4.A.3 Associated Legendre Functions.- 4.A.4 Generalized Laguerre Polynomials.- 4.A.5 Surface Measure on the Sphere and Gamma Function.- 4.A.6 Green Function of the Laplacian.- 5 Regularization and Symmetry.- 5.1 Moser Method.- 5.2 Souriau Method.- 5.2.1 Fock Parameters.- 5.2.2 Bacry-Györgyi Parameters.- 5.3 Kustaanheimo-Stiefel Transformation.- II Group-Geometric Theory 109.- 6 Conformal Regularization.- 6.1 The Conformal Group.- 6.2 The Compactified Minkowski Space.- 6.3 The Cotangent Bundle to Minkowski Space.- 6.4 Regularization of the Kepler Problem.- 7 Spinorial Regularization.- 7.1 The Homomorphism SU(2, 2) ? SO(2, 4).- 7.1.1 Two Bases for su(2, 2).- 7.1.2 SU(2, 2) and Compactified Minkowski Space.- 7.2 Return to the Kustaanheimo-Stiefel Map.- 7.3 Generalized Kustaanheimo-Stiefel Map.- 8 Return to Separation of Variables.- 8.1 Separable Orthogonal Systems.- 8.1.1 Stäckel Theorem.- 8.1.2 Eisenhart Theorem.- 8.1.3 Robertson Theorem.- 8.2 Finding Coordinate Systems Separating Kepler Problem.- 8.2.1 Spherical Coordinates.- 8.2.2 Parabolic Coordinates.- 8.2.3 Elliptic Coordinates.- 8.2.4 Spheroconical Coordinates.- 8.3 Integrable Perturbations.- 8.3.1 Euler Problem.- 8.3.2 Stark Problem.- Append.- 8.A Jacobian Elliptic Functions.- 9 Geometric Quantization.- 9.1 Multiplier Representations.- 9.2 Quantization of Geodesics on the Sphere.- 9.3 Quantization of the Kepler Problem.- 10 Kepler Problem with Magnetic Monopole.- 10.1 Nonnull Twistors and Magnetic Monopoles.- 10.1.1 Bound Motions.- 10.1.2 Unbound Motions.- 10.1.3 Quantization.- 10.2 The MICZ System.- 10.3 The Taub-NUT System.- 10.4 The BPST Instanton.- III Perturbation Theory 235.- 11 General Perturbation Theory.- 11.1 Formal Expansions.- 11.1.1 Lie Series and Formal Canonical Transformations.- 11.1.2 Homological Equation and its Formal Solution.- 11.2 The Convergence Problem.- 11.2.1 Convergence of Lie Series.- 11.2.2 Homological Equation and its Solution.- 11.2.3 Kolmogorov Theorem.- 11.2.4 Nekhoroshev Theorem.- Appendices.- 11.AResults from Diophantine Theory.- 11.B Cauchy Inequality.- 12 Perturbations of the Kepler Problem.- 12.1 A More Convenient Hamiltonian.- 12.2 Normalization (or Averaging) Method.- 12.3 Numerical Integration.- 12.3.1 Symbolic Manipulation.- 12.3.2 Compiling Equations.- Appendices.- 12.AVariation of the Constants.- 12.B The Stabilization Method.- 13 Perturbations with Axial Symmetry.- 13.1 Reduction of Orbit Manifold.- 13.2 Lunar Problem.- 13.3 Stark and Quadratic Zeeman Effect.- 13.4 Satellite around Oblate Primary.- IV Appendices 321.- A Differential Geometry.- A.1 Rudiments of Topology.- A.2 Differentiable Manifolds.- A.2.1 Definition.- A.2.2 Tangent and Cotangent Spaces.- A.2.3 Push-forward and Pull-back.- A.3 Tensors and Forms.- A.3.1 Tensors.- A.3.2 Forms and Exterior Derivatives.- A.3.3 Lie Derivative.- A.3.4 Integration of Differential Forms.- A.4 Distributions and Frobenius Theorem.- A.5 Riemannian, Symplectic and Poisson Manifolds.- A.5.1 Riemannian Manifolds.- A.5.2 Symplectic Manifolds.- A.5.3 Poisson Manifolds.- A.6 Fibre Bundles.- A.6.1 Definition.- A.6.2 Principal and Associated Fibre Bundles.- B Lie Groups and Lie Algebras.- B.1 Definition and Properties.- B.2 Adjoint and Coadjoint Representation.- B.3 Action of a Lie Group on a Manifold.- B.4 Classification of Lie Groups and Lie Algebras.- B.5 Connection on a Principal Bundle.- C Lagrangian Dynamics.- C.1 Lagrange Equations.- C.2 Hamilton Principle.- C.3 Noether Theorem.- C.4 Reduced Lagrangian and Maupertuis Principle.- D Hamiltonian Dynamics.- D.1 From Lagrange to Hamilton.- D.2 The Hamilton-Jacobi Integration Method.- D.2.1 Canonical Transformations.- D.2.2 Hamilton-Jacobi Eq…