The Kepler Problem: Group Theoretical Aspects, Regularization and Quantization, with Application to the Study of Pertur by Bruno CordaniThe Kepler Problem: Group Theoretical Aspects, Regularization and Quantization, with Application to the Study of Pertur by Bruno Cordani

The Kepler Problem: Group Theoretical Aspects, Regularization and Quantization, with Application to…

byBruno Cordani

Paperback | October 3, 2013

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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.
Title:The Kepler Problem: Group Theoretical Aspects, Regularization and Quantization, with Application to…Format:PaperbackDimensions:442 pagesPublished:October 3, 2013Publisher:Springer-Verlag/Sci-Tech/TradeLanguage:English

The following ISBNs are associated with this title:

ISBN - 10:303489421X

ISBN - 13:9783034894210

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Table of Contents

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 <_20_0.-20_2.520_the20_repulsive20_potential.-20_append.-20_2.a20_the20_kepler20_equation.-20_320_separation20_of20_variables20_and20_action-angle20_coordinates.-20_3.120_separation20_of20_variables.-20_3.1.120_spherical20_coordinates.-20_3.1.220_parabolic20_coordinates.-20_3.1.320_elliptic20_coordinates.-20_3.1.420_spheroconical20_coordinates.-20_3.220_action-angle20_variables.-20_3.2.120_delaunay20_and20_poincarc3a9_20_variables.-20_3.2.220_pauli20_variables.-20_3.2.320_monodromy.-20_420_quantization20_of20_the20_kepler20_problem.-20_4.120_the20_schrc3b6_dinger20_quantization.-20_4.1.120_spherical20_coordinates.-20_4.1.220_parabolic20_coordinates.-20_4.1.320_elliptic20_coordinates.-20_4.1.420_spheroconical20_coordinates.-20_4.220_pauli20_quantization.-20_4.2.120_canonical20_quantization.-20_4.2.220_pauli20_quantization.-20_4.320_fock20_quantization.-20_append.-20_4.a20_mathematical20_review.-20_4.a.120_second20_order20_linear20_differential20_equations.-20_4.a.220_laplacian20_on20_the20_sphere20_and20_homogeneous20_harmonic20_polynomials.-20_4.a.320_associated20_legendre20_functions.-20_4.a.420_generalized20_laguerre20_polynomials.-20_4.a.520_surface20_measure20_on20_the20_sphere20_and20_gamma20_function.-20_4.a.620_green20_function20_of20_the20_laplacian.-20_520_regularization20_and20_symmetry.-20_5.120_moser20_method.-20_5.220_souriau20_method.-20_5.2.120_fock20_parameters.-20_5.2.220_bacry-gyc3b6_rgyi20_parameters.-20_5.320_kustaanheimo-stiefel20_transformation.-20_ii20_group-geometric20_theory20_109.-20_620_conformal20_regularization.-20_6.120_the20_conformal20_group.-20_6.220_the20_compactified20_minkowski20_space.-20_6.320_the20_cotangent20_bundle20_to20_minkowski20_space.-20_6.420_regularization20_of20_the20_kepler20_problem.-20_720_spinorial20_regularization.-20_7.120_the20_homomorphism20_su28_22c_20_229_20_3f_20_so28_22c_20_429_.-20_7.1.120_two20_bases20_for20_su28_22c_20_229_.-20_7.1.220_su28_22c_20_229_20_and20_compactified20_minkowski20_space.-20_7.220_return20_to20_the20_kustaanheimo-stiefel20_map.-20_7.320_generalized20_kustaanheimo-stiefel20_map.-20_820_return20_to20_separation20_of20_variables.-20_8.120_separable20_orthogonal20_systems.-20_8.1.120_stc3a4_ckel20_theorem.-20_8.1.220_eisenhart20_theorem.-20_8.1.320_robertson20_theorem.-20_8.220_finding20_coordinate20_systems20_separating20_kepler20_problem.-20_8.2.120_spherical20_coordinates.-20_8.2.220_parabolic20_coordinates.-20_8.2.320_elliptic20_coordinates.-20_8.2.420_spheroconical20_coordinates.-20_8.320_integrable20_perturbations.-20_8.3.120_euler20_problem.-20_8.3.220_stark20_problem.-20_append.-20_8.a20_jacobian20_elliptic20_functions.-20_920_geometric20_quantization.-20_9.120_multiplier20_representations.-20_9.220_quantization20_of20_geodesics20_on20_the20_sphere.-20_9.320_quantization20_of20_the20_kepler20_problem.-20_1020_kepler20_problem20_with20_magnetic20_monopole.-20_10.120_nonnull20_twistors20_and20_magnetic20_monopoles.-20_10.1.120_bound20_motions.-20_10.1.220_unbound20_motions.-20_10.1.320_quantization.-20_10.220_the20_micz20_system.-20_10.320_the20_taub-nut20_system.-20_10.420_the20_bpst20_instanton.-20_iii20_perturbation20_theory20_235.-20_1120_general20_perturbation20_theory.-20_11.120_formal20_expansions.-20_11.1.120_lie20_series20_and20_formal20_canonical20_transformations.-20_11.1.220_homological20_equation20_and20_its20_formal20_solution.-20_11.220_the20_convergence20_problem.-20_11.2.120_convergence20_of20_lie20_series.-20_11.2.220_homological20_equation20_and20_its20_solution.-20_11.2.320_kolmogorov20_theorem.-20_11.2.420_nekhoroshev20_theorem.-20_appendices.-20_11.aresults20_from20_diophantine20_theory.-20_11.b20_cauchy20_inequality.-20_1220_perturbations20_of20_the20_kepler20_problem.-20_12.120_a20_more20_convenient20_hamiltonian.-20_12.220_normalization20_28_or20_averaging29_20_method.-20_12.320_numerical20_integration.-20_12.3.120_symbolic20_manipulation.-20_12.3.220_compiling20_equations.-20_appendices.-20_12.avariation20_of20_the20_constants.-20_12.b20_the20_stabilization20_method.-20_1320_perturbations20_with20_axial20_symmetry.-20_13.120_reduction20_of20_orbit20_manifold.-20_13.220_lunar20_problem.-20_13.320_stark20_and20_quadratic20_zeeman20_effect.-20_13.420_satellite20_around20_oblate20_primary.-20_iv20_appendices20_321.-20_a20_differential20_geometry.-20_a.120_rudiments20_of20_topology.-20_a.220_differentiable20_manifolds.-20_a.2.120_definition.-20_a.2.220_tangent20_and20_cotangent20_spaces.-20_a.2.320_push-forward20_and20_pull-back.-20_a.320_tensors20_and20_forms.-20_a.3.120_tensors.-20_a.3.220_forms20_and20_exterior20_derivatives.-20_a.3.320_lie20_derivative.-20_a.3.420_integration20_of20_differential20_forms.-20_a.420_distributions20_and20_frobenius20_theorem.-20_a.520_riemannian2c_20_symplectic20_and20_poisson20_manifolds.-20_a.5.120_riemannian20_manifolds.-20_a.5.220_symplectic20_manifolds.-20_a.5.320_poisson20_manifolds.-20_a.620_fibre20_bundles.-20_a.6.120_definition.-20_a.6.220_principal20_and20_associated20_fibre20_bundles.-20_b20_lie20_groups20_and20_lie20_algebras.-20_b.120_definition20_and20_properties.-20_b.220_adjoint20_and20_coadjoint20_representation.-20_b.320_action20_of20_a20_lie20_group20_on20_a20_manifold.-20_b.420_classification20_of20_lie20_groups20_and20_lie20_algebras.-20_b.520_connection20_on20_a20_principal20_bundle.-20_c20_lagrangian20_dynamics.-20_c.120_lagrange20_equations.-20_c.220_hamilton20_principle.-20_c.320_noether20_theorem.-20_c.420_reduced20_lagrangian20_and20_maupertuis20_principle.-20_d20_hamiltonian20_dynamics.-20_d.120_from20_lagrange20_to20_hamilton.-20_d.220_the20_hamilton-jacobi20_integration20_method.-20_d.2.120_canonical20_transformations.-20_d.2.220_hamilton-jacobi20_equation.-20_d.2.320_geometric20_description.-20_d.2.420_the20_time-dependent20_case.-20_d.320_symmetries20_and20_reduction.-20_d.3.120_the20_moment20_map.-20_d.3.220_reduction20_of20_symplectic20_manifolds.-20_d.3.320_reduction20_of20_poisson20_manifolds.-20_d.420_action-angle20_variables.-20_d.4.120_arnold20_theorem.-20_d.4.220_degenerate20_systems.-20_d.4.320_monodromy. 0.-="" 2.5="" the="" repulsive="" potential.-="" append.-="" 2.a="" kepler="" equation.-="" 3="" separation="" of="" variables="" and="" action-angle="" coordinates.-="" 3.1="" variables.-="" 3.1.1="" spherical="" 3.1.2="" parabolic="" 3.1.3="" elliptic="" 3.1.4="" spheroconical="" 3.2="" 3.2.1="" delaunay="" _poincarc3a9_="" 3.2.2="" pauli="" 3.2.3="" monodromy.-="" 4="" quantization="" problem.-="" 4.1="" _schrc3b6_dinger="" quantization.-="" 4.1.1="" 4.1.2="" 4.1.3="" 4.1.4="" 4.2="" 4.2.1="" canonical="" 4.2.2="" 4.3="" fock="" 4.a="" mathematical="" review.-="" 4.a.1="" second="" order="" linear="" differential="" equations.-="" 4.a.2="" laplacian="" on="" sphere="" homogeneous="" harmonic="" polynomials.-="" 4.a.3="" associated="" legendre="" functions.-="" 4.a.4="" generalized="" laguerre="" 4.a.5="" surface="" measure="" gamma="" function.-="" 4.a.6="" green="" function="" laplacian.-="" 5="" regularization="" symmetry.-="" 5.1="" moser="" method.-="" 5.2="" souriau="" 5.2.1="" parameters.-="" 5.2.2="" _bacry-gyc3b6_rgyi="" 5.3="" kustaanheimo-stiefel="" transformation.-="" ii="" group-geometric="" theory="" 109.-="" 6="" conformal="" regularization.-="" 6.1="" group.-="" 6.2="" compactified="" minkowski="" space.-="" 6.3="" cotangent="" bundle="" to="" 6.4="" 7="" spinorial="" 7.1="" homomorphism="" _su28_22c_="" _229_="" _so28_22c_="" _429_.-="" 7.1.1="" two="" bases="" for="" _229_.-="" 7.1.2="" 7.2="" return="" map.-="" 7.3="" 8="" 8.1="" separable="" orthogonal="" systems.-="" 8.1.1="" _stc3a4_ckel="" theorem.-="" 8.1.2="" eisenhart="" 8.1.3="" robertson="" 8.2="" finding="" coordinate="" systems="" separating="" 8.2.1="" 8.2.2="" 8.2.3="" 8.2.4="" 8.3="" integrable="" perturbations.-="" 8.3.1="" euler="" 8.3.2="" stark="" 8.a="" jacobian="" 9="" geometric="" 9.1="" multiplier="" representations.-="" 9.2="" geodesics="" sphere.-="" 9.3="" 10="" problem="" with="" magnetic="" monopole.-="" 10.1="" nonnull="" twistors="" monopoles.-="" 10.1.1="" bound="" motions.-="" 10.1.2="" unbound="" 10.1.3="" 10.2="" micz="" system.-="" 10.3="" taub-nut="" 10.4="" bpst="" instanton.-="" iii="" perturbation="" 235.-="" 11="" general="" theory.-="" 11.1="" formal="" expansions.-="" 11.1.1="" lie="" series="" transformations.-="" 11.1.2="" homological="" equation="" its="" solution.-="" 11.2="" convergence="" 11.2.1="" series.-="" 11.2.2="" 11.2.3="" kolmogorov="" 11.2.4="" nekhoroshev="" appendices.-="" 11.aresults="" from="" diophantine="" 11.b="" cauchy="" inequality.-="" 12="" perturbations="" 12.1="" a="" more="" convenient="" hamiltonian.-="" 12.2="" normalization="" _28_or="" _averaging29_="" 12.3="" numerical="" integration.-="" 12.3.1="" symbolic="" manipulation.-="" 12.3.2="" compiling="" 12.avariation="" constants.-="" 12.b="" stabilization="" 13="" axial="" 13.1="" reduction="" orbit="" manifold.-="" 13.2="" lunar="" 13.3="" quadratic="" zeeman="" effect.-="" 13.4="" satellite="" around="" oblate="" primary.-="" iv="" appendices="" 321.-="" geometry.-="" a.1="" rudiments="" topology.-="" a.2="" differentiable="" manifolds.-="" a.2.1="" definition.-="" a.2.2="" tangent="" spaces.-="" a.2.3="" push-forward="" pull-back.-="" a.3="" tensors="" forms.-="" a.3.1="" tensors.-="" a.3.2="" forms="" exterior="" derivatives.-="" a.3.3="" derivative.-="" a.3.4="" integration="" a.4="" distributions="" frobenius="" a.5="" _riemannian2c_="" symplectic="" poisson="" a.5.1="" riemannian="" a.5.2="" a.5.3="" a.6="" fibre="" bundles.-="" a.6.1="" a.6.2="" principal="" b="" groups="" algebras.-="" b.1="" definition="" properties.-="" b.2="" adjoint="" coadjoint="" representation.-="" b.3="" action="" group="" b.4="" classification="" b.5="" connection="" bundle.-="" c="" lagrangian="" dynamics.-="" c.1="" lagrange="" c.2="" hamilton="" principle.-="" c.3="" noether="" c.4="" reduced="" maupertuis="" d="" hamiltonian="" d.1="" hamilton.-="" d.2="" hamilton-jacobi="" d.2.1="" d.2.2="" d.2.3="" description.-="" d.2.4="" time-dependent="" case.-="" d.3="" symmetries="" reduction.-="" d.3.1="" moment="" d.3.2="" d.3.3="" d.4="" d.4.1="" arnold="" d.4.2="" degenerate="" d.4.3="">

Editorial Reviews

"This is an interesting book, which well organizes the group-geometric aspects of the Kepler problem on which a great number of articles have been published along with the advance of symmetry theory. . . . a nice reference not only for graduate students but also for scientists who are interested in dynamical systems with symmetry." --MathSciNet