# Classical Dynamics of Particles and Systems

## 1st Edition

**Authors:**Jerry B. Marion

**eBook ISBN:**9781483272818

**Imprint:**Academic Press

**Published Date:**1st January 1965

**Page Count:**592

## Description

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

Problems

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

Problems

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

Problems

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

Problems

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

Problems

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

Problems

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

Problems

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

Problems

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

Problems

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

Problems

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

Problems

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

Problems

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

Problems

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

Problems

Solutions, Hints, and References for Selected Problems

Appendix A. Taylor's Theorem

Exercises

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

Exercises

Appendix C. Ordinary Differential Equations of Second Order

C.1 Linear Homogeneous Equations

C.2 Linear Inhomogeneous Equations

Exercises

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

Bibliography

## Details

- No. of pages:
- 592

- Language:
- English

- Copyright:
- © Academic Press 1965

- Published:
- 1st January 1965

- Imprint:
- Academic Press

- eBook ISBN:
- 9781483272818