Classical Dynamics of Particles and Systems

Classical Dynamics of Particles and Systems

1st Edition - January 1, 1965

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  • Author: Jerry B. Marion
  • eBook ISBN: 9781483272818

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Classical Dynamics of Particles and Systems presents a modern and reasonably complete account of the classical mechanics of particles, systems of particles, and rigid bodies for physics students at the advanced undergraduate level. The book aims to present a modern treatment of classical mechanical systems in such a way that the transition to the quantum theory of physics can be made with the least possible difficulty; to acquaint the student with new mathematical techniques and provide sufficient practice in solving problems; and to impart to the student some degree of sophistication in handling both the formalism of the theory and the operational technique of problem solving. Vector methods are developed in the first two chapters and are used throughout the book. Other chapters cover the fundamentals of Newtonian mechanics, the special theory of relativity, gravitational attraction and potentials, oscillatory motion, Lagrangian and Hamiltonian dynamics, central-force motion, two-particle collisions, and the wave equation.

Table of Contents

  • Preface

    Chapter 1. Matrices and Vectors

    1.1 Introduction

    1.2 The Concept of a Scalar

    1.3 Coordinate Transformations

    1.4 Properties of Rotation Matrices

    1.5 Matrix Operations

    1.6 Further Definitions

    1.7 Geometrical Significance of Transformation Matrices

    1.8 Definitions of a Scalar and a Vector in Terms of Transformation Properties

    1.9 Elementary Scalar and Vector Operations

    1.10 The Scalar Product of Two Vectors

    1.11 The Vector Product of Two Vectors

    1.12 Unit Vectors

    Suggested References


    Chapter 2. Vector Calculus

    2.1 Introduction

    2.2 Differentiation of a Vector with Respect to a Scalar

    2.3 Examples of Derivatives —Velocity and Acceleration

    2.4 Angular Velocity

    2.5 The Gradient Operator

    2.6 The Divergence of a Vector

    2.7 The Curl of a Vector

    2.8 Some Additional Differential Vector Relations

    2.9 Integration of Vectors

    Suggested References


    Chapter 3. Fundamentals of Newtonian Mechanics

    3.1 Introduction

    3.2 Newton's Laws

    3.3 Frames of Reference

    3.4 The Equation of Motion for a Particle

    3.5 Conservation Theorems

    3.6 Conservation Theorems for a System of Particles

    3.7 Limitations of Newtonian Mechanics

    Suggested References


    Chapter 4. The Special Theory of Relativity

    4.1 Introduction

    4.2 Galilean Invariance

    4.3 The Lorentz Transformation

    4.4 Momentum and Energy in Relativity

    4.5 Some Consequences of the Lorentz Transformation

    Suggested References


    Chapter 5. Gravitational Attraction and Potentials

    5.1 Introduction

    5.2 The Gravitational Potential

    5.3 Lines of Force and Equipotential Surfaces

    5.4 The Gravitational Potential of a Spherical Shell

    5.5 A Final Comment

    Suggested References


    Chapter 6. Oscillatory Motion

    6.1 Introduction

    6.2 The Simple Harmonic Oscillator

    6.3 Damped Harmonic Motion

    6.4 Forcing Functions

    6.5 Forced Oscillations

    6.6 Phase Diagrams

    6.7 The Response of Linear Oscillators to Impulsive Forcing Functions

    6.8 Electrical Oscillations

    6.9 Harmonic Oscillations in Two Dimensions

    6.10 The Use of Complex Notation

    Suggested References

    Problems 7

    Chapter 7. Nonlinear Oscillations

    7.1 Oscillations

    7.2 Oscillations for General Potential Functions

    7.3 Phase Diagrams for Nonlinear Systems

    7.4 The Plane Pendulum

    7.5 Nonlinear Oscillations in a Symmetric Potential - The Method of Successive Approximations

    7.6 Nonlinear Oscillations in an Asymmetric Potential - The Method of Perturbations

    Suggested References


    Chapter 8. Some Methods in the Calculus of Variations

    8.1 Introduction

    8.2 Statement of the Problem

    8.3 Euler's Equation

    8.4 The Brachistochrone Problem

    8.5 The "Second Form" of Euler's Equation

    8.6 Functions with Several Dependent Variables

    8.7 The Euler Equations When Auxiliary Conditions Are Imposed

    8.8 The δ Notation

    Suggested References


    Chapter 9. Hamilton's Principle — Lagrangian and Hamiltonian Dynamics

    9.1 Introduction

    9.2 Hamilton's Principle

    9.3 Generalized Coordinates

    9.4 Lagrange's Equations of Motion in Generalized Coordinates

    9.5 Lagrange's Equations with Undetermined Multipliers

    9.6 The Equivalence of Lagrange's and Newton's Equations

    9.7 The Essence of Lagrangian Dynamics

    9.8 A Theorem Concerning the Kinetic Energy

    9.9 The Conservation of Energy

    9.10 The Conservation of Linear Momentum

    9.11 The Conservation of Angular Momentum

    9.12 The Canonical Equations of Motion — Hamiltonian Dynamics

    9.13 Some Comments Regarding Dynamical Variables and Variational Calculations in Physics

    9.14 Phase Space and Liouville's Theorem

    9.15 The Virial Theorem

    9.16 The Lagrangian Function in Special Relativity

    Suggested References


    Chapter 10. Central-Force Motion

    10.1 Introduction

    10.2 The Reduced Mass

    10.3 Conservation Theorems — First Integrals of the Motion

    10.4 Equations of Motion

    10.5 Orbits in a Central Field

    10.6 Centrifugal Energy and the Effective Potential

    10.7 Planetary Motion-Kepler's Problem

    10.8 Kepler's Equation

    10.9 Approximate Solution of Kepler's Equation

    10.10 Apsidal Angles and Precession

    10.11 Stability of Circular Orbits

    10.12 The Problem of Three Bodies

    Suggested References


    Chapter 11. Kinematics of Two-Particle Collisions

    11.1 Introduction

    11.2 Elastic Collisions —Center-of-Mass and Laboratory Coordinate Systems

    11.3 Kinematics of Elastic Collisions

    11.4 Cross Sections

    11.5 The Rutherford Scattering Formula

    11.6 The Total Cross Section

    11.7 Relativistic Kinematics

    Suggested References


    Chapter 12. Motion in a Noninertial Reference Frame

    12.1 Introduction

    12.2 Rotating Coordinate Systems

    12.3 The Coriolis Force

    12.4 Motion Relative to the Earth

    Suggested References


    Chapter 13. Dynamics of Rigid Bodies

    13.1 Introduction

    13.2 The Inertia Tensor

    13.3 Angular Momentum

    13.4 Principal Axes of Inertia

    13.5 Moments of Inertia for Different Body Coordinate Systems

    13.6 Further Properties of the Inertia Tensor

    13.7 The Eulerian Angles

    13.8 Euler's Equations for a Rigid Body

    13.9 Force-Free Motion of a Symmetrical Top

    13.10 The Motion of a Symmetrical Top with One Point Fixed

    13.11 The Stability of Rigid-Body Rotations

    Suggested References


    Chapter 14. Systems with Many Degrees of Freedom — Small Oscillations and Normal Coordinates

    14.1 Introduction

    14.2 Two Coupled Harmonic Oscillators

    14.3 The General Problem of Coupled Oscillations

    14.4 The Orthogonality of the Eigenvectors

    14.5 Normal Coordinates

    14.6 Two Linearly Coupled Plane Pendula

    14.7 Three Linearly Coupled Plane Pendula — An Example of Degeneracy

    14.8 The Loaded String

    14.9 The Continuous String as a Limiting Case of the Loaded String

    14.10 The Wave Equation

    14.11 The Nonuniform String - Orthogonal Functions and Perturbation Theory

    14.12 Fourier Analysis

    Suggested References


    Chapter 15. The Wave Equation in One Dimension

    15.1 Introduction

    15.2 Separation of the Wave Equation

    15.3 Phase Velocity, Dispersion, and Attenuation

    15.4 Electrical Analogies — Filtering Networks

    15.5 Group Velocity and Wave Packets

    15.6 Fourier Integral Representation of Wave Packets

    15.7 Energy Propagation in the Loaded String

    15.8 Further Comments Regarding Phase and Group Velocities

    15.9 Reflected and Transmitted Waves

    15.10 Damped Plane Waves

    Suggested References


    Solutions, Hints, and References for Selected Problems

    Appendix A. Taylor's Theorem


    Appendix B. Complex Numbers

    B.1 Complex Numbers

    B.2 Geometrical Representation of Complex Numbers

    B.3 Trigonometric Functions of Complex Variables

    B.4 Hyperbolic Functions


    Appendix C. Ordinary Differential Equations of Second Order

    C.1 Linear Homogeneous Equations

    C.2 Linear Inhomogeneous Equations


    Appendix D. Useful Formulas

    D.1 Binomial Expansion

    D.2 Trigonometric Relations

    D.3 Trigonometric Series

    D.4 Exponential and Logarithmic Series

    D.5 Hyperbolic Functions

    Appendix E. Useful Integrals

    E.1 Algebraic Functions

    E.2 Trigonometric Functions

    E.3 Gamma Functions

    E.4 Elliptic Integrals

    Appendix F. Differential Relations in Curvilinear Coordinate Systems

    F.1 Cylindrical Coordinates

    F.2 Spherical Coordinates

    Appendix G. A Proof of the Relation Σµ χ 2µ = Σµ χ' 2µ

    Selected References


Product details

  • No. of pages: 592
  • Language: English
  • Copyright: © Academic Press 1965
  • Published: January 1, 1965
  • Imprint: Academic Press
  • eBook ISBN: 9781483272818

About the Author

Jerry B. Marion

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