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Our presentation of some fundamental domains of celestial mechanics requires no special preliminary knowledge; however, the chosen mathe matical method is new in so far as the pure two-body motion is described by linear differential equations, which even have constant coefficients. In other words an equivalence between the Keplerian motion and a harmonic oscillation is established; this approach to celestial mechanics will be referred to as the linear theory. Besides the possibility of the mutual fruitful interaction between celestial and oscillatory mechanics which is thereby created, our linear differential equations are as a result everywhere regular. The opposite is true of the classical Newtonian equations, which are singular at the collision of the two moving bodies"'Reg~larization is however not the leitmotiv of the book; the many regularization methods [1] which do not lead to linear differential equations are therefore not described. Apart from the basic idea of the use of linear differential equations and the resulting advantages, there were two further scientific goals which we had in mind. First, it should be permissible not only to transform the coordinates of the mobile but also to introduce other independent variables instead of the time. The often cumbersome solution of the Keplerian equation in theoretical studies can thereby be avoided. This leads to the further consequence that the linear theory is uniform with respect to the value of the eccentricity.

Linear and Regular Celestial Mechanics: Perturbed Two-body Motion Numerical Methods Canonical Theory

I Basic and Numerical Theory.- I Preliminaries.- 1. Laws of Motion.- Motion about a Central Mass.- Perturbed Motion.- 2. Energy Relations.- Perturbing Potential.- 3. Collection of Formulae.- 4. Singular Differential Equations.- 5. One-Dimensional Motion.- First Step of Regularization.- Second Step of Regularization.- The Harmonic Oscillator.- II Regularized Theory.- 6. Matrices.- 7. Fictitious lime.- 8. Motion in a Plane.- 9. Motion in Space.- The KS-Matrix.- Equations of Motion.- Energy Relations.- Regularized Equations.- Comments.- Further Aids.- Collection of Formulae.- III Kepler Motion.- 10. Elliptic Motion.- Geometric Properties.- Dynamics of Elliptical Motion.- 11. Uniform Treatment of the Three Varieties of Pure Kepler Motion.- Stumpff Functions.- Kepler Motion in Terms of the c-Functions.- Collection of Formulae.- IV The Initial Value Problem.- 12. Shape and Position of the Orbit.- 13. Use of the KS-Transformation.- Pure Motion in Parametric Representation.- The Parametric Initial Value Problem.- The Osculating Orbit.- Collection of Formulae.- The Problem of the Projectile.- Critical Initial Data.- 14. The Classical Elements.- Collection of Formulae.- 15. Numerical Examples.- V The Fundamental Differential Equations.- 16. Stability.- Comments.- 17. Numerical Aspects.- Regulation of the Step Size.- Error Propagation.- 18. The Time-Element.- The General Concept of Elements.- of a Time-Element.- Collection of Formulae.- Comments.- 19. A Set of Regular Elements.- Collection of Formulae.- Comments.- The u-Method.- Appendix. Orthogonal Elements.- VI Typical Perturbations.- 20. Gravitational Potentials.- Clusters and Rigid Bodies.- Speroids.- Example.- 21. Oblateness Perturbations.- The r-Equation.- Comments.- 22. Third Body Attraction.- The Interior Problem.- Rule for Adjustment of the Perturbing Potential (Interior Problem).- The Exterior Problem.- Comments.- 23. Numerical Examples.- Adaptation of the Element Equations (19,61)-(19,63) to Automatic Computation.- General Description of the Experiments.- Examples 1-6.- VII Refmed Numerical Methods.- 24. Difference Methods.- Derivation of the Modified Coefficients.- Collection of Formulae.- 25. Applications to Differential Equations.- Application to Celestial Mechanics.- Long Term Numerical Behaviour.- 26. Refinement of Taylor Expansions.- Definitions and Properties of a Sequence of Functions.- On Solving Differential Equations by Power Expansions.- Finite Power- and G-Expansions.- Algorithm.- Remarks.- 27. Application to Satellite Motion.- Oblateness Perturbation.- Comments.- Third Body Attraction.- Final Remark Concerning Sections 24 to 27.- 28. First Order Methods.- Third Body Perturbation.- Comments.- Convergence of Fourier Expansions.- Expansion with Respect to a Perturbing Parameter.- II Canonical Theory.- VIII General Canonical Theory.- 29. Preliminaries.- Dynamics of n Point Masses.- General Canonical Systems.- 30. Homogeneous Formalism.- Law of Energy.- Homogeneous Systems.- 31. Canonical Transformations.- 32. Generating Function.- Independent Variable Appearing Explicitly in a Transformation.- 33. Construction of Canonical Transformations.- Transformation of the Coordinates.- Jacobi's Method of Integration.- Elements and their Perturbations.- 34. New Independent Variable, Scaling Factors.- IX Classical Canonical Theory of the Perturbations of Elements.- 35. Delaunay Elements.- Spherical Coordinates.- Jacobi's Integration Method.- Canonical and Classical Elements.- Delaunay Elements.- 36. Perturbation of Elements.- Perturbation of the Delaunay Elements.- Perturbation of the Classical Elements.- Secular Differential Equations.- X The Canonical Theory of the Oscillator Generated by the Perturbed Problem of two Bodies.- 37. Introduction of new Independent Variables.- 38. The Canonical KS-Transformation.- Motion in Space, Canonical KS-Transformation.- Fundamental Theorem.- The Basic Canonical System with Respect to the Fictitious Time s.- The Basic Canonical System with Respect to the Generalized Eccentric Anomaly E.- 39. Canonical Forces.- The System in the Fictitious Time s.- The System in the Generalized Eccentric Anomaly E.- 40. Canonical Oscillator Elements in the Fictitious Time Version.- Separation of Jacobi's Equation.- Uniform Treatment of the Element Equations.- Uniformly Regular Canonical Elements.- The Basic Element System with Respect to the Fictitious Time s.- 41. Canonical Oscillator Elements in the Generalized Eccentric Anomaly Version.- The Basic System with Respect to the Generalized Eccentric Anomaly E.- Canonical Forces.- 42. First Views on Analytical Perturbation Methods.- Literal Expression of the Perturbing Potential as a Function of the Canonical Oscillator Elements.- Analytical First Order Perturbations.- Comments Concerning Chapters IX and X.- III Geometry and Outlook.- XI Geometry of the KS-Transformation.- 43. The Fourdimensional Space.- The Fibration of U4.- The Mapping of Curves.- Planes of Levi-Civita's Type.- Remarks.- The Cayley Matrix.- A Cross Product.- Remarks.- 44. The Mapping of the Spheres.- The Fibration of S3.- The Inverse Mapping of Curves.- Quaternions.- Topological Aids.- 45. The Manifold of Kepler Orbits.- Outlook.- References.- Author and Subject Index.