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New Foundations for Classical Mechanics (Second Edition) by David Hestenes

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New Foundations for Classical Mechanics (Second Edition) written by David Hestenes , Department of Physics and Astronomy, Arizona State University, Tempe, Arizona, U.S.A. The second edition has been expanded by nearly a hundred pages on relativistic mechanics. The treatment is unique in its exclusive use of geometric algebra and its detailed treatment of spacetime maps, collisions, motion in uniform fields and relativistic spin precession. It conforms with Einstein’s view that Special Relativity is the culmination of developments in classical mechanics.
The accuracy of the text has been improved by the accumulation of many corrections over the last decade. I am grateful to the many students and colleagues who have helped root out errors, as well as the invaluable assistance of Patrick Reany in preparing the manuscript. The second edition, in particular, has benefited from careful scrutiny by J. L. Jones and Prof. J. Vrbik. The most significant corrections are to the perturbation calculations in Chapter 8. Prof. Vrbik located the error in my calculation of the precession of the moon's orbit due to perturbation by the sun (p. 550), a calculation which vexed Newton and many others since. I am indebted to David Drewer for calling my attention to D.T. Whiteside’s fascinating account of Newton’s failure to master the lunar perigee calculation (see Section 8-3). Vrbik has kindly contributed a more accurate computation to this edition. He has also extended the spinor perturbation theory of Section 8-4 in a series of published applications to celestial mechanics (see References). Unfortunately, to make room for the long relativity chapter, the chapter on Foundations of Mechanics had to be dropped from the Second Edition. It will be worth expanding at another time. Indeed, it has already been incorporated in a new appraoch to physics instruction centered on making and using conceptual models. [For an update on Modeling Theory, see D. Hestenes, “Modeling Games in the Newtonian World,” Am. J. Phys. 60, 732–748 (1992).]
When using this book as a mechanics textbook, it is important to move quickly through Chapters 1 and 2 to the applications in Chapter 3. A thorough study of the topics and problems in Chapter 2 could easily take the better part of a semester, so that chapter should be used mainly for reference in a mechanics course. To facilitate identification of those elements of geometric algebra which are most essential to applications, a Synopsis of Geometric Algebra has been included in the beginning of this edition.

New Foundations for Classical Mechanics (Second Edition) written by David Hestenes cover the following topics.

  • Preface ix

  • 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 v3: 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
    Chapter 9: Relativistic Mechanics
    9-1.Spacetime and Its Representations
    9-2.Spacetime Maps and Measurements
    9-3.Relativistic Particle Dynamics
    9-4.Energy-Momentum Conservation
    9-5.Relativistic Rigid Body Mechanics

  • Appendix
    A Spherical Trigonometry
    B Elliptic Functions
    C Units, Constants And Data
    Hints and Solutions for Selected Exercises

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