New Foundations for Classical Mechanics(revised) This is a textbook on classical mechanics at the intermediate level, but its main purpose is to serve as an introduction to a new mathematical language for physics called geometric algebra. Mechanics is most commonly formulated today in terms of the vector algebra developed by the American physicist J. Willard Gibbs, but for some applications of mechanics the algebra of complex numbers is more efficient than vector algebra, while in other applications matrix algebra works better. Geometric algebra integrates all these algebraic systems into a coherent mathematical language which not only retains the advantages of each special algebra but possesses powerful new capabilities. This book covers the fairly standard material for a course on the mechanics of particles and rigid bodies. However, it will be seen that geometric algebra brings new insights into the treatment of nearly every topic and produces simplifications that move the subject quickly to advanced levels. That has made it possible in this book to carry the treatment of two major topics in mechanics well beyond the level of other textbooks. A few words are in order about the unique treatment of these two topics, namely, rotational dynamics and celestial mechanics. |
Contents
Developments in Geometric Algebra | 39 |
Mechanics of a Single Particle | 120 |
Central Forces and TwoParticle Systems | 195 |
Operators and Transformations | 252 |
ManyParticle Systems | 334 |
Rigid Body Mechanics | 419 |
Celestial Mechanics | 512 |
Relativistic Mechanics | 574 |
Appendix | 661 |
Hints and Solutions for Selected Exercises | 674 |
| 690 | |
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Common terms and phrases
angle angular momentum approximation atomic bivector calculate called center of mass component compute conservation constant of motion coordinates corresponding curve defined derivative describes determined differential direction displacement Earth effect eigenvalues eigenvector Einstein’s electromagnetic elliptical equation of motion equivalent Euler expressed factor force law formula frame frequency function geometric algebra geometric product given harmonic implies inertia tensor inertial system integral interpretation Kepler kinetic energy line segments linear operator Lorentz transformation magnetic field magnitude mathematical matrix mechanics multiplication multivector Newton’s Newtonian normal modes Note obtain orbit orthogonal oscillator outer products parameters parametric equation particle pendulum perturbation physical plane position potential precession problem properties Prove pseudoscalar radius reduces relation relative result rigid body rotational velocity satellite scalar scattering Section Show shown in Figure solution solve space spacetime specified sphere spherical spin spinor spinor equation surface symmetry axis theorem theory torque trajectory triangle unit vector variables write