New Foundations for Classical Mechanics

1: Origins of Geometric Algebra.- 1-1. Geometry as Physics.- 1-2. Number and Magnitude.- 1-3. Directed Numbers.- 1-4. The Inner Product.- 1-5. The Outer Product.- 1-6. Synthesis and Simplification.- 1-7. Axioms for Geometric Algebra.- 2: Developments in Geometric Algebra.- 2-1. Basic Identities and Definitions.- 2-2. The Algebra of a Euclidean Plane.- 2-3. The Algebra of Euclidean 3-Space.- 2-4. Directions, Projections and Angles.- 2-5. The Exponential Function.- 2-6. Analytic Geometry.- 2-7. Functions of a Scalar Variable.- 2-8. Directional Derivatives and Line Integrals.- 3: Mechanics of a Single Particle.- 3-1. Newton's Program.- 3-2. Constant Force.- 3-3. Constant Force with Linear Drag.- 3-4. Constant Force with Quadratic Drag.- 3-5. Fluid Resistance.- 3-6. Constant Magnetic Field.- 3-7. Uniform Electric and Magnetic Fields.- 3-8. Linear Binding Force.- 3-9. Forced Oscillations.- 3-10. Conservative Forces and Constraints.- 4: Central Forces and Two-Particle Systems.- 4-1. Angular Momentum.- 4-2. Dynamics from Kinematics.- 4-3. The Kepler Problem.- 4-4. The Orbit in Time.- 4-5. Conservative Central Forces.- 4-6. Two-particle Systems.- 4-7. Elastic Collisions.- 4-8. Scattering Cross Sections.- 5: Operators and Transformations.- 5-1. Linear Operators and Matrices.- 5-2. Symmetric and Skewsymmetric Operators.- 5-3. The Arithmetic of Reflections and Rotations.- 5-4. Transformation Groups.- 5-5. Rigid Motions and Frames of Reference.- 5-6. Motion in Rotating Systems.- 6: Many-Particle Systems.- 6-1. General Properties of Many-Particle Systems.- 6-2. The Method of Lagrange.- 6-3. Coupled Oscillations and Waves.- 6-4. Theory of Small Oscillations.- 6-5. The Newtonian Many Body Problem.- 7: Rigid Body Mechanics.- 7-1. Rigid Body Modeling.- 7-2. Rigid Body Structure.- 7-3. The Symmetrical Top.- 7-4. Integrable Cases of Rotational Motion.- 7-5. Rolling Motion.- 7-6. Impulsive Motion.- 8: Celestial Mechanics.- 8-1. Gravitational Forces, Fields and Torques.- 8-2. Perturbations of Kepler Motion.- 8-3. Perturbations in the Solar System.- 8-4. Spinor Mechanics and Perturbation Theory.- 9: Foundations of Mechanics.- 9-1. Models and Theories.- 9-2. The Zeroth Law of Physics.- 9-3. Generic Laws and Principles of Particle Mechanics.- 9-4. Modeling Processes.- Appendixes.- A Spherical Trigonometry.- B Elliptic Functions.- C Units, Constants and Data.- Hints and Solutions for Selected Exercises.- References.