Foundations of Celestial Mechanics

Introduction
References
Contents
1 About the N-Body Problem
1.1 Self-Gravitating Systems of Massive Points
1.2 The Fundamental First Integrals
1.2.1 Conservation of momentum
1.2.2 The Conservation of Angular Momentum
1.2.3 The Conservation of Energy
1.3 The N-Body Problem in Barycentric and in Relative Systems
1.4 On the Solution of the N-Body Problem
1.5 The Virial Theorem: Classical Formulation
References
2 On the Two-Body Problem
2.1 Motion Relative to the Center of Mass
2.2 Reduction to the Plane
2.3 The Effective Potential Energy
2.4 The Trajectory
2.5 Laplace-Runge-Lenz Vector
2.6 Geometry of Conic Orbits
2.6.1 Ellipse
2.6.2 Parabola
2.6.3 Hyperbola
2.7 Properties of Conic Orbits
2.7.1 Elliptical Orbit (calE<0)
2.7.2 Average Velocity on an Elliptical Orbit
2.7.3 Parabolic Orbit (calE=0)
2.7.4 Hyperbolic Orbit (calE>0)
2.8 Keplerian Elements of the Orbit
2.9 Calculation of the Ephemerides
2.10 The Laplace Method for Recovering the Orbital Elements from Observations
2.11 Application to Ballistics and Space Flight
References
3 The Three-Body Problem
3.1 Stationary Solutions
3.1.1 Collinear Solutions
3.1.2 Triangular Solutions
3.1.3 Stationary Solutions: Summary
3.2 The Circular Restricted Three-Body Problem
3.3 The Zero-Velocity Curves
3.3.1 The (x,y) Plane
3.3.2 The (x,z) Plane
3.3.3 The (y,z) Plane
3.4 Considerations on the Lagrangian Points
3.4.1 Case of Dominant Mass
3.5 Stability of the Lagrangian Points
3.5.1 Conditions for the Equilibrium
3.5.2 Collinear solutions
3.5.3 Triangular Solution: Points L4 and L5
3.6 The Variation of the Elements
3.6.1 Variation of the Orientation Elements
3.6.2 Variation of the Geometric Elements
References
4 Analytical Mechanics
4.1 The Function of Lagrange
4.2 Lagrange Function in Generalized Coordinates
4.3 The Equations of Lagrange
4.4 The Hamilton Function
4.5 Some Considerations on the Canonical Equations
4.6 Constants of Motion, Poisson and Lagrange Brackets
4.7 Lagrange and Poisson Brackets for the Elliptical Orbit
4.8 Canonical Transformations
4.8.1 Characteristic Function
4.8.2 The Various Forms of the Characteristic Function
4.8.3 Conditions for Canonicity Using the Poisson and Lagrange Brackets
4.8.4 The Canonical Invariants
4.8.5 The Infinitesimal Canonical Transformations
4.8.6 Canonical Systems of Constants of Motion
4.8.7 A System of Canonical Elements for the Elliptical Orbit
4.9 The Equation of Jacobi
4.10 Special Cases of the Jacobi Equation
4.10.1 When the Hamiltonian does not Depend on Time
4.10.2 Separation of Variables
4.11 The Method of Hamilton-Jacobi Applied to the Two-Body Problem
4.12 The Variation of Elements
4.12.1 Example to Illustrate the Method of the Variation of Constants
4.13 Apsidal Precession
4.14 The Two-Body Problem in General Relativity
References
5 Gravitational Potential
5.1 The Theorem of Gauss or of the Flux
5.2 The Equations of Poisson and Laplace
5.3 The Potential of a Massive Point
5.4 The Potential of Spherical Bodies
5.5 The Equation of Legendre
5.5.1 Spherical Harmonics
5.5.2 Legendre Equation and Spherical Harmonics
5.5.3 Particular Solution of the Legendre Equation: the Associated Functions
5.5.4 The Spherical Harmonics of Integer Degree
5.6 Expansion of the Potential through Spherical Harmonics
5.7 Potential of a Thin Spherical Layer
5.8 Potential of a Homogeneous Spheroid in an External Point
5.9 Potential of a Homogeneous Ellipsoid in an Inner Point
5.10 Potential of a Homogeneous Ellipsoid in an External Point
5.11 The Explicit Form of the Potential in an External Point
5.12 Rotational Distortion of the Earth
5.13 Potential of a Homogeneous Circular Torus
5.13.1 Potential of an Infinitely Thin Ring
5.13.2 Potential of a Homogeneous Circular Torus
5.13.3 Approximation of the Homogeneous Torus Potential in the Outer Region
References
Appendix A Fundamental Formulas of Spherical Trigonometry
Appendix B Formulas of Transformation of Coordinate Systems
Appendix C About Vector Operators
Appendix D Relative Motion
Appendix E The Mirror Theorem
Appendix F On the Solution of Kepler's Equation
F.1 Lagrange's Inversion Theorem
F.2 Fourier's Expansion Theorem
F.3 Numerical Solutions: An Iterative Method
Appendix G Bohr's Model for the Hydrogen Atom
G.1 Bohr's Model of the Hydrogen Atom
Appendix H The Variation of Constants
Appendix I Lagrange Multipliers
Appendix J The Two-Body Problem Applied to Visual Binary Orbits
Appendix K Planarity of the Stationary Solution of the 3-Body Problem
Appendix L A Simplified Approach to Gravitational Impact
Appendix M The Brackets of Poisson and Lagrange
M.1 Poisson Brackets
M.2 Lagrange Brackets
M.3 Relations Between the Brackets of Poisson and Lagrange
Appendix N Some Special Functions
N.1 Gamma Function
N.2 Beta Function
N.3 Bessel Functions
N.3.1 Bessel Functions of the First Kind
N.3.2 Bessel Functions of the Second Kind
N.3.3 Modified Bessel Functions
N.4 Elliptical Integrals
Appendix O Orthogonal Functions
O.1 Least Squares
O.2 A Family of Orthogonal Polynomials
O.3 Legendre Polynomials
O.4 Spherical Harmonics
O.4.1 Harmonics Representation
O.4.2 Orthogonality of Spherical Harmonics
O.4.3 Addition Theorem
O.5 An Application of Spherical Harmonics
Appendix P Some Theorems on Harmonic Functions
P.1 Selected Theorems
P.2 Special Problems
Appendix Q On the Principles of Analytical Mechanics
Q.1 Variational Formulation of Motion
Q.2 Conservation Laws
Q.3 Maupertuis's Principle
Q.4 The Geodesic Lines
Appendix R Space-Time Invariant and Conservation Principles
R.1 Continuous Trajectories
R.2 Time-Invariance and Conservation of Energy
R.3 Invariance to Translations and Conservation of Moment
R.4 Rotational Invariance and Angular Momentum Conservation
References
Index