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.
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Developments in Geometric Algebra
Mechanics of a Single Particle
Central Forces and TwoParticle Systems
Rigid Body Mechanics
93 94 95 Relativistic Particle Dynamics EnergyMomentum Conservation Relativistic Rigid Body Mechanics 615 633 650
Appendix A Spherical Trigonometry B Elliptic Functions C Units Constants and Data Hints and Solutions for Selected Exercises References Index
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angle angular momentum approximation atomic bivector body problem calculate called center of mass central force circular component compute conservation constants of motion coordinates corresponding defined derivative describes determined differential direction Earth effect eigenvalues eigenvector elliptical equation of motion equilibrium equivalent Euler expressed factor force law formulation frame frequency function geometric algebra geometric product given gravitational harmonic inertia tensor inertial system initial conditions integral interpretation Kepler’s kinetic energy line segments linear operator linear transformation magnetic field magnitude mathematical matrix elements mechanics multivector normal modes Note obtain orbit orthogonal oscillator outer products parameters parametric equation particle pendulum perturbation physical plane position precession principal vectors properties proved radius reduces relation relative result rigid body rotational motion rotational velocity scalar scattering Section Show shown in Figure solution solve space spacetime specified sphere spinning spinor spinor equation surface symmetry axis theorem theory torque trajectory triangle unit vector variables write