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.- 2.2.2 Energy H > 0.- 2.2.3 Energy H = 0.- 2.3 The Three Anomalies.- 2.3.1 Energy H < 0.- 2.3.2 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 P
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.
