Classical and Celestial Mechanics

Name: Classical and Celestial Mechanics | The Recife Lectures
Brand: Princeton University Press
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The Recife Lectures

Hildeberto Cabral Florin Diacu(Herausgeber*in)

Princeton University Press

1. Auflage

Erschienen am 8. Dezember 2020

408 Seiten

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Inhalt

Cover Page
Half-title Page
Title Page
Copyright Page
Contents
Foreword
Preface
Central Configurations and Relative Equilibria for the JV-Body Problem
1. Introduction
2. Dziobek's Coordinates
3. Configurations for the Three-Body Problem
4. Configurations in the Four-Body Problem
5. Configurations in the Five-Body Problem
6. Palmore's Coordinates
Appendix A: The Area of a Triangle
Appendix B: Mathematica Code for the Four-Body Problem
Appendix C: Mathematica Code for the Planar Five-Body Problem
References
Singularities of the TV-Body Problem
1. Introduction
2. First Integrals
3. Singularities
4. Collisions
5. Pseudocollisions
6. Particular Cases
7. Clustered Configurations
8. Examples of Pseudocollisions
9. Relationships between Singularities
10. Extensions beyond Collision
11. Block Regularization
12. The Collision Manifold
13. The Case
14. The Case
15. The Case
16. Conclusions and Perspectives
References
Lectures on the Two-Body Problem
Introduction
Lecture 1. Preliminaries
Lecture 2. Two Solutions by Reduction
Lecture 3. Why Are Keplerian Orbits Closed?
Lecture 4. Concerning the Eccentricity Vector
Lecture 5. Lambert's Theorem
Bibliography and Author Index
Normal Forms of Hamiltonian Systems and Stability of Equilibria
1. Introduction
2. Hamiltonian Systems
3. Symplectic Changes of Coordinates and Generating Functions
4. Hamiltonian Flows
5. Stability of Equilibria
6. Normal Forms
7. The Linear Normalization
8. Some Stability Results
9. The Restricted Three-Body Problem
10. Deprit-Hori's Normalization Scheme.Proof of Theorem 6.1
References
Poincare's Compactification and Applications to Celestial Mechanics
1. Introduction
2. Poincare Compactification for Polynomial Vector Fields
3. Poincare Compactincation for Polynomial Hamiltonian Vector Fields
4. Generic Properties
5. Behavior at Infinity in the Monomial Case
6. The Kepler Problem
6.1 The Kepler Problem on the Line
6.2 The Kepler Problem in the Plane
7. The Poincare Compactification for Homogeneous Functions
8. The Kepler Problem without Regularization
8.1 The Kepler Problem on the Line
8.2 The Kepler Problem in the Plane
9. Hill's Problem
References
The Motion of the Moon
1. Remarks about the Accuracy of the Solution
2. The Equations of Motion
3. The Solution Method
4. The Intermediate Orbit
5. The Terms of First Order in the Inclination
6. Terms at First Order in e
Appendix A: Canonical Transformation to Jacobi Coordinates
Appendix B: MACSYMA Program for the Intermediate Orbit
Appendix C: MACSYMA Program for Inclination
Appendix D: MACSYMA Program for First Order Terms in e
References
Lectures on Geometrical Methods in Mechanics
1. Variational Principles:Their Explanation and Applications
1.1 Explanation of Fermat's Principle
1.2 High Frequency Asymptotics: From Waves to Rays
1.3 Solving First-Order PDEs Using an Optical Analogy
1.4 From Optics to Mechanics by Bernoulli
1.5 The Maupertuis-Euler-Lagrange Principle
1.6 Hamiltonian Mechanics via Optical Motivation
2. Geometrical Basics of Optimal Control
2.1 Heuristics of the Maximum Principle
2.2 A Geometric Derivation of the Euler-Lagrange Equations
3. Geometric Phases in Mechanics
3.1 A Shopping Cart Problem
3.2 The Holonomy of a Bike Wheel
3.3 Dual Cones
3.4 Proof of the Gauss-Bonnet Formula Using Dual Cones
3.5 Duality of External and of Internal Billiards on the Sphere
3.6 Geometric Phases in the Motion of Free Rigid Bodies
3.7 Euler's Equations of Motion: An Instantaneous Derivation
3.8 Fluid in a Tube
3.9 A Spherimeter
3.10 Optical Fibers
References
Momentum Maps and Geometric Phases
Foreword
Part 1: Overview
1. Adiabatic Phases: Overview
2. Nonadiabatic Phases of Gyrostats
3. Holonomy with Zero Momentum
References
Part 2: Classical Adiabatic Angles
4. Averaging Heuristics
5. The Classical Adiabatic Phases
6. Pendulum with Slowly Varying Gravity
7. Slowly Moving Integrable Mechanical Systems
8. Hannay-Berry's 1-Form for Moving Systems
9. The Main Theorem
9.1 Invariance Under a Maximal Torus Tr
9.2 Isotropic Oscillator
9.3 Foucault's Pendulum Revisited
10. Final Remarks
References
Part 3: Holonomy for Gyrostats
11. The Hamiltonian
12. Derivation of Formula (14)
12.1 Where and What to Integrate?
12.2 Applying Stokes's Theorem
12.3 Line Integral over Cx
12.4 Line Integral over C2
12.5 The Surface Integral over E
13. Holonomy for the Gyrostat
14. Final Remarks
15. Derivation of the Hamiltonian
References
Part 4: Microswimming
16. Historical Remarks
17. The Configuration Space
18. Hydrodynamics
19. The Momentum Mapping
20. Swimming=Holonomy
21. Nonspherical Self-Reparametrizing Cells
21.1 Prolate and Oblate Spheroids
21.2 Outline of the Calculation
21.3 Results for Prolate and Oblate Spheroids
21.4 Comparisons of Swimming Velocities
22. Final Remarks
References
Bifurcation from Families of Periodic Solutions
1. Introduction
2. The Alternative Method
3. Bifurcation near Families of Solutions
3.1 Interaction of Forcing and Damping in a Nonlinear Oscillator
3.2 The Predator-Prey Model of Lotka-Volterra with Periodic Forcing
References
Index

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