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Intro; Preface; Introduction; Contents; Symbols; 1 Elements of Vector Calculus; 1.1 Vector Functions of Real Variables; 1.2 Limits of Vector Functions; 1.3 Derivatives of Vector Functions; 1.3.1 Geometric Interpretation; 1.4 Integration of Vector Functions; 1.5 The Formal Vector Operator; 1.5.1 Gradient and in Spherical Polar Coordinates; 1.5.2 Gradient and in Cylindrical Coordinates; 1.6 The Divergence Operator; 1.7 The Curl; 1.8 Formal Calculation of Divergence and Laplacian by the Operator; 1.8.1 Spherical Polar Coordinates; 1.8.2 Cylindrical Coordinates; 1.9 Vector Fields
1.9.1 Lines of Force of a Vector Field1.10 The Divergence Theorem; 1.10.1 The Stokes' Theorem; 1.11 Velocity Fields; 1.12 Meaning of the Divergence of a Vector Field; 1.13 Link Between Divergence and Volume Variation; 1.14 Solenoidal Vector Fields; 1.14.1 Flux Tubes; 1.15 Further Readings; 2 Newtonian Gravitational Interaction; 2.1 Single Particle Gravitational Potential; 2.2 Motion of a Particle Gravitating in a Resisting Medium; 2.3 The Gravitational N-Body Case; 2.3.1 Potential of N Gravitating Bodies; 2.3.2 Mechanical Energy of the N Bodies; 2.4 The Scalar Virial Theorem
2.4.1 Consequences of the Virial Theorem2.5 Continuous Distributions of Matter; 2.5.1 Poisson's and Laplace's Equations; 2.6 Gauss' Theorem; 2.7 Gravitational Potential Energy; 2.8 Newton's Theorems; 2.9 Further Readings; 3 Central Force Fields; 3.1 The Potential and Force Generated by a Spherical Matter Distribution; 3.1.1 Calculating Spherical Potentials via Poisson's Equation; 3.2 Motion in a Spherical Potential; 3.2.1 Circular Trajectories; 3.3 Potential Generated by a Homogeneous Sphere; 3.3.1 Trajectories in a Homogeneous Sphere; 3.3.2 Radial Motion in a Homogeneous Sphere
3.4 Some Relevant Spherical Models3.4.1 The Plummer Sphere; 3.4.2 The Isochrone Potential; 3.5 Quality of Motion; 3.5.1 The Keplerian Case; 3.5.2 The Homogeneous Sphere Case; 3.6 Periods of Oscillations; 3.6.1 Radial Period in the Keplerian Potential; 3.6.2 Radial Period in the Homogeneous Sphere Potential; 3.6.3 Radial Period in the Plummer Potential; 3.6.4 Radial Period in the Isochrone Potential; 3.7 Azimuthal Period; 3.7.1 Fully Periodic Motion; 3.8 The Inverse Problem in a Central Force Field; 3.8.1 From Elliptic Trajectories to their Central Force Field; 3.9 Further Readings
1.9.1 Lines of Force of a Vector Field1.10 The Divergence Theorem; 1.10.1 The Stokes' Theorem; 1.11 Velocity Fields; 1.12 Meaning of the Divergence of a Vector Field; 1.13 Link Between Divergence and Volume Variation; 1.14 Solenoidal Vector Fields; 1.14.1 Flux Tubes; 1.15 Further Readings; 2 Newtonian Gravitational Interaction; 2.1 Single Particle Gravitational Potential; 2.2 Motion of a Particle Gravitating in a Resisting Medium; 2.3 The Gravitational N-Body Case; 2.3.1 Potential of N Gravitating Bodies; 2.3.2 Mechanical Energy of the N Bodies; 2.4 The Scalar Virial Theorem
2.4.1 Consequences of the Virial Theorem2.5 Continuous Distributions of Matter; 2.5.1 Poisson's and Laplace's Equations; 2.6 Gauss' Theorem; 2.7 Gravitational Potential Energy; 2.8 Newton's Theorems; 2.9 Further Readings; 3 Central Force Fields; 3.1 The Potential and Force Generated by a Spherical Matter Distribution; 3.1.1 Calculating Spherical Potentials via Poisson's Equation; 3.2 Motion in a Spherical Potential; 3.2.1 Circular Trajectories; 3.3 Potential Generated by a Homogeneous Sphere; 3.3.1 Trajectories in a Homogeneous Sphere; 3.3.2 Radial Motion in a Homogeneous Sphere
3.4 Some Relevant Spherical Models3.4.1 The Plummer Sphere; 3.4.2 The Isochrone Potential; 3.5 Quality of Motion; 3.5.1 The Keplerian Case; 3.5.2 The Homogeneous Sphere Case; 3.6 Periods of Oscillations; 3.6.1 Radial Period in the Keplerian Potential; 3.6.2 Radial Period in the Homogeneous Sphere Potential; 3.6.3 Radial Period in the Plummer Potential; 3.6.4 Radial Period in the Isochrone Potential; 3.7 Azimuthal Period; 3.7.1 Fully Periodic Motion; 3.8 The Inverse Problem in a Central Force Field; 3.8.1 From Elliptic Trajectories to their Central Force Field; 3.9 Further Readings