# Classical Mechanics

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## Book Description

**Classical Mechanics, Second Edition** presents a complete account of the classical mechanics of particles and systems for physics students at the advanced undergraduate level. The book evolved from a set of lecture notes for a course on the subject taught by the author at California State University, Stanislaus, for many years. It assumes the reader has been exposed to a course in calculus and a calculus-based general physics course. However, no prior knowledge of differential equations is required. Differential equations and new mathematical methods are developed in the text as the occasion demands.

The book begins by describing fundamental concepts, such as velocity and acceleration, upon which subsequent chapters build. The second edition has been updated with two new sections added to the chapter on Hamiltonian formulations, and the chapter on collisions and scattering has been rewritten. The book also contains three new chapters covering Newtonian gravity, the Hamilton-Jacobi theory of dynamics, and an introduction to Lagrangian and Hamiltonian formulations for continuous systems and classical fields. To help students develop more familiarity with Lagrangian and Hamiltonian formulations, these essential methods are introduced relatively early in the text.

The topics discussed emphasize a modern perspective, with special note given to concepts that were instrumental in the development of modern physics, for example, the relationship between symmetries and the laws of conservation. Applications to other branches of physics are also included wherever possible. The author provides detailed mathematical manipulations, while limiting the inclusion of the more lengthy and tedious ones. Each chapter contains homework problems of varying degrees of difficulty to enhance understanding of the material in the text. This edition also contains four new appendices on D'Alembert's principle and Lagrange's equations, derivation of Hamilton’s principle, Noether’s theorem, and conic sections.

## Table of Contents

**Kinematics: Describing the Motion**Introduction

Space, Time, and Coordinate Systems

Change of Coordinate System (Transformation of Components of a Vector)

Displacement Vector

Speed and Velocity

Acceleration

Velocity and Acceleration in Polar Coordinates

Angular Velocity and Angular Acceleration

Infinitesimal Rotations and the Angular Velocity Vector

**Newtonian Mechanics**

The First Law of Motion (Law of Inertia)

The Second Law of Motion; the Equations of Motion

The Third Law of Motion

Galilean Transformations and Galilean Invariance

Newton’s Laws of Rotational Motion

Work, Energy, and Conservation Laws

Systems of Particles

References

**Integration of Newton’s Equation of Motion**

Introduction

Motion Under Constant Force

Force Is a Function of Time

Force Is a Function of Velocity

Force Is a Function of Position

Time-Varying Mass System (Rocket System)

**Lagrangian Formulation of Mechanics: Descriptions of Motion in Configuration Space**

Generalized Coordinates and Constraints

Kinetic Energy in Generalized Coordinates

Generalized Momentum

Lagrangian Equations of Motion

Nonuniqueness of the Lagrangian

Integrals of Motion and Conservation Laws

Scale Invariance

Nonconservative Systems and Generalized Potential

Charged Particle in Electromagnetic Field

Forces of Constraint and Lagrange’s Multipliers

Lagrangian versus Newtonian Approach to Classical Mechanics

Reference

**Hamiltonian Formulation of Mechanics: Descriptions of Motion in PhaseSpaces**

The Hamiltonian of a Dynamic System

Hamilton’s Equations of Motion

Integrals of Motion and Conservation Theorems

Canonical Transformations

Poisson Brackets

Poisson Brackets and Quantum Mechanics

Phase Space and Liouville’s Theorem

Time Reversal in Mechanics (Optional)

Passage from Hamiltonian to Lagrangian

References

**Motion Under a Central Force**

Two-Body Problem and Reduced Mass

General Properties of Central Force Motion

Effective Potential and Classification of Orbits

General Solutions of Central Force Problem

Inverse Square Law of Force

Kepler’s Three Laws of Planetary Motion

Applications of Central Force Motion

Newton’s Law of Gravity from Kepler’s Laws

Stability of Circular Orbits (Optional)

Apsides and Advance of Perihelion (Optional)

Laplace–Runge–Lenz Vector and the Kepler Orbit (Optional)

References

**Harmonic Oscillator**

Simple Harmonic Oscillator

Adiabatic Invariants and Quantum Condition

Damped Harmonic Oscillator

Phase Diagram for Damped Oscillator

Relaxation Time Phenomena

Forced Oscillations without Damping

Forced Oscillations with Damping

Oscillator Under Arbitrary Periodic Force

Vibration Isolation

Parametric Excitation

**Coupled Oscillations and Normal Coordinates**

Coupled Pendulum

Coupled Oscillators and Normal Modes: General Analytic Approach

Forced Oscillations of Coupled Oscillators

Coupled Electric Circuits

**Nonlinear Oscillations**

Qualitative Analysis: Energy and Phase Diagrams

Elliptical Integrals and Nonlinear Oscillations

Fourier Series Expansions

The Method of Perturbation

Ritz Method

Method of Successive Approximation

Multiple Solutions and Jumps

Chaotic Oscillations

References

**Collisions and Scatterings**

Direct Impact of Two Particles

Scattering Cross Sections and Rutherford Scattering

Laboratory and Center-of-Mass Frames of Reference

Nuclear Sizes

Small-Angle Scattering (Optional)

References

**Motion in Non-Inertial Systems**

Accelerated Translational Coordinate System

Dynamics in Rotating Coordinate System

Motion of Particle Near the Surface of the Earth

Foucault Pendulum

Larmor’s Theorem

Classical Zeeman Effect

Principle of Equivalence

**Motion of Rigid Bodies**

Independent Coordinates of Rigid Body

Eulerian Angles

Rate of Change of Vector

Rotational Kinetic Energy and Angular Momentum

Inertia Tensor

Euler’s Equations of Motion

Motion of a Torque-Free Symmetrical Top

Motion of Heavy Symmetrical Top with One Point Fixed

Stability of Rotational Motion

References

**Theory of Special Relativity**

Historical Origin of Special Theory of Relativity

Michelson–Morley Experiment

Postulates of Special Theory of Relativity

Lorentz Transformations

Doppler Effect

Relativistic Space–Time (Minkowski Space)

Equivalence of Mass and Energy

Conservation Laws of Energy and Momentum

Generalization of Newton’s Equation of Motion

Relativistic Lagrangian and Hamiltonian Functions

Relativistic Kinematics of Collisions

Collision Threshold Energies

References

**Newtonian Gravity and Newtonian Cosmology**

Newton’s Law of Gravity

Gravitational Field and Gravitational Potential

Gravitational Field Equations: Poisson’s and Laplace’s Equations

Gravitational Field and Potential of Extended Body

Tides

General Theory of Relativity: Relativistic Theory of Gravitation

Introduction to Cosmology

Brief History of Cosmological Ideas

Discovery of Expansion of the Universe, Hubble’s Law

Big Bang

Formulating Dynamical Models of the Universe

Cosmological Red Shift and Hubble Constant

*H*

Critical Mass Density and Future of the Universe

Microwave Background Radiation

Dark Matter

Reference

**Hamilton–Jacobi Theory of Dynamics**

Canonical Transformation and H-J Equation

Action and Angle Variables

Infinitesimal Canonical Transformations and Time Development Operator

H-J Theory and Wave Mechanics

Reference

**Introduction to Lagrangian and Hamiltonian Formulations for Continuous Systems and Classical Fields**

Vibration of Loaded String

Vibrating Strings and the Wave Equation

Continuous Systems and Classical Fields

Scalar and Vector of Fields

**Appendix 1: Vector Analysis and Ordinary Differential Equations**

Appendix 2: D’Alembert’s Principle and Lagrange’s Equations

Appendix 3: Derivation of Hamilton’s Principle from D’Alembert’s Principle

Appendix 4: Noether’s Theorem

Appendix 5: Conic Sections, Ellipse, Parabola, and Hyperbola

Index

Appendix 2: D’Alembert’s Principle and Lagrange’s Equations

Appendix 3: Derivation of Hamilton’s Principle from D’Alembert’s Principle

Appendix 4: Noether’s Theorem

Appendix 5: Conic Sections, Ellipse, Parabola, and Hyperbola

Index

## Author(s)

### Biography

**Dr. Tai Chow** was born and raised in China. He received the Bachelor of Science degree in physics from National Taiwan University, a master’s degree in physics from Case Western Reserve University in Cleveland, and a Ph.D. degree in physics from the University of Rochester in New York. Since 1970, Dr. Chow has been in the Department of Physics at California State University, Stanislaus, and served as the department chairman for 18 years. He has published more than 40 articles in physics and astrophysics journals and is the author of four textbooks.