The Kepler Problem

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 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 Equation.- D.2.3 Geometric Description.- D.2.4 The Time-dependent Case.- D.3 Symmetries and Reduction.- D.3.1 The Moment Map.- D.3.2 Reduction of Symplectic Manifolds.- D.3.3 Reduction of Poisson Manifolds.- D.4 Action-Angle Variables.- D.4.1 Arnold Theorem.- D.4.2 Degenerate Systems.- D.4.3 Monodromy.