After the launching of the first artificial satellites preceding interplanetary vehicles, celestial mechanics is no longer a science of interest confined to a small group of astronomers and mathematicians; it becomes a special engineering technique. I have tried to set this book in this new perspective, by severely limiting the choice of examples from classical celestial mechanics and by retaining only those useful in calculating the trajectory of a body in space. The main chapter in this book is the fifth, where a detailed solution is given of the problem of motion of an artificial satellite in the Earth's gravitational field, using the methods of Von Zeipel and of Brouwer. It is shown how Lagrange's equations can be applied to this problem. The first four chapters contain proofs of the main results useful for these two methods: the elliptical solution of the two-body problem and the basic algebra of celestial mechanics; some theorems of analytical mechanics; the Delaunay variables and the Lagrangian equations of variation of elements; the expansion of the disturbing function and the Bessel functions necessary for this expansion. The last two chapters are more descriptive in character. In them I have summarized briefly some of the classical theories of celestial mechanics, and have tried to show their distinctive characteristics without going into details.

I / The Principles of Celestial Mechanics.- 1. The Fundamental Law of General Mechanics.- 2. General Theorems of Mechanics.- 3. Newton's Law.- 4. The Scope and Limitations of Newton's Law.- 5. The TV-Body Problem.- 6. Equations of the N-Body Problem.- 7. Integrals of the N-Body Problem.- II / The Two-Body Problem.- 8. The Importance of the Two-Body Problem.- 9. Absolute and Relative Motion of Two Bodies.- 10. Form of the Trajectories.- 11. Kepler's Law.- 12. Study of Elliptical Motion.- 13. The Orbital Elements.- 14. Cartesian Coordinates of the Body.- 15. Astronomical Units in the Solar System.- III / Systems of Canonical Equations.- 16. TV-Body Problem Equations in a Relative Frame of Reference.- 17. Reductions of the Equations for a Three-Body Problem.- 18. The Case in which One of the Bodies has Negligible Mass.- 19. Canonical Form of the Equations.- 20. The Case in which F is not a Function of t.- 21. Integral of a System of Canonical Equations.- 22. Canonical Transformations of Variables.- 23. Examples of Canonical Transformations.- 24. Jacobi's Theorem.- 25. Canonical Equations for the Two-Body Problem.- 26. Application of Jacobi's Theorem to the Two-Body Problem.- 27. Meaning of the Constants a.- 28. Variables Conjugate to the Qi.- 29. Application of these Results to the General Problem.- 30. The Delaunay Variables.- 31. Osculating Elements.- 32. Lagrange Equations.- 33. The Case of Zero Eccentricity or Zero Inclination.- IV / Perturbation Theory.- 34. Introduction.- 35. Fourier Series.- 36. Expansion of the Eccentric Anomaly in Fourier Series.- 37. Definition of Bessel Functions.- 38. Some Properties of Bessel Functions.- 39. Expansion of cos jE and sin jE.- 40. Expression of Other Functions of the Two-Body Problem.- 41. Relation between E and v.- 42.D'Alembert's Property.- 43. Limited Expansions in e.- 44. Convergence of Series Expanded in Powers of e.- 45. Expression of the Disturbing Function (the Case of the Moon).- 46. Reduction to the Variables of Elliptical Motion.- 47. Expansion of the Disturbing Function.- 48. Expansion in a Small Parameter.- 49. Theorems of Proof.- 50. Form of Equations with the Osculating Elements.- 51. Method of Solution.- 52. Long-Period and Short-Period Terms.- 53. Convergence of the Series of the Solution.- V / The Motion of an Artificial Satellite.- 54. The Potential of a Rigid Body.- 55. Expansion of the Potential.- 56. The Case of a Nearly Spherical Body.- 57. Equations of Motion of an Artificial Satellite.- 58. The Principle of Von Zeipel's Method.- 59. Setting Up of the Equations.- 60. Elimination of the Mean Anomaly.- 61. Explicit Expression for S1.- 62. Calculation of ?'2.- 63. Elimination of g.- 64. Main Results: the Motion of Artificial Satellites.- 65. Application of the Lagrange Equations; First Approximation.- 66. Second Approximation with the Lagrange Equations.- 67. Comparison of the Two Methods.- 68. The Case of Very Small Eccentricity and Inclination.- 69. Critical Inclination.- 70. Libration of the Perigee near the Critical Inclination.- 71. The Phenomenon of Libration.- VI / Lunar Theory and the Motion of the Satellites.- 72. The Principal Problem of the Lunar Theory.- 73. Approximate Solution of the Main Problem.- 74. The Principal Inequalities of the Motion of the Moon.- 75. Various Lunar Theories.- 76. Delaunay's Theory.- 77. The Theory of Hill and Brown.- 78. Hansen's Theory.- 79. Improvement of the Theories.- 80. Problems of the Motion of Other Natural Satellites.- VII / The Planetary Theory.- 81. The Disturbing Function.- 82. First-OrderSolution.- 83. Expansion of the Disturbing Function by Harmonic Analysis.- 84. Other Numerical Expansions.- 85. Perturbations by Forces in Rectangular Coordinates.- 86. Variables in Hansen's Method.- 87. Calculation by Hansen's Method.- 88. Higher-Order Planetary Theories.- 89. Purely Numerical Methods.- 90. The Form of Numerical Integration.- 91. Starting the Numerical Integration.- 92. The Numerical Integration Proper.- 93. Properties of Numerical Integration.- 94. The Use of Numerical Integration.- 95. Comparison between Numerical Integration and Analytical Theories.