Classical Mechanics

631 Pages·2013·18.3 MB·English

by  Tai L. Chow

MECHANICS

Classical Mechanics Classical Mechanics

Second Edition

Second Edition

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
Jupiter
at California State University, Stanislaus, for many years. It assumes the reader has been
orbit
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
Earth
methods are developed in the text as the occasion demands.
orbit

The book begins by describing fundamental concepts, such as velocity and acceleration,
Sun
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
Initial
scattering has been rewritten. The book also contains three new chapters covering Newtonian
spacecraft
gravity, the Hamilton-Jacobi theory of dynamics, and an introduction to Lagrangian and
orbit
Hamiltonian formulations for continuous systems and classical fields. To help students develop
more familiarity with Lagrangian and Hamiltonian formulations, these essential methods
Final
are introduced relatively early in the text.
spacecraft
The topics discussed emphasize a modern perspective, with special note given to concepts orbit
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 Outer
homework problems of varying degrees of difficulty to enhance understanding of the planet
orbit
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.
Second
Edition
Tai L. Chow
K16463
ISBN: 978-1-4665-6998-0
90000
9 781466 569980
K16463_Cover_mech.indd All Pages 3/20/13 10:12 AM

Classical Mechanics
Second Edition

Classical Mechanics
Second Edition
Tai L. Chow
Boca Raton London New York
CRC Press is an imprint of the
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Version Date: 20130227
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Contents
Preface..............................................................................................................................................xv
Author ............................................................................................................................................xvii
Chapter 1 Kinematics: Describing the Motion .............................................................................1
1.1 Introduction .......................................................................................................1
1.2 Space, Time, and Coordinate Systems ..............................................................1
1.3 Change of Coordinate System (Transformation of Components of a
Vector) ........................................................................................................3
1.4 Displacement Vector ..........................................................................................8
1.5 Speed and Velocity ............................................................................................8
1.6 Acceleration .....................................................................................................10
1.6.1 Tangential and Normal Acceleration..................................................11
1.7 Velocity and Acceleration in Polar Coordinates .............................................14
1.7.1 Plane Polar Coordinates (r, θ) ............................................................14
1.7.2 Cylindrical Coordinates (ρ, ϕ, z) ........................................................15
(cid:31)
1.7.3 Spherical Coordinates (r, θ, ϕ) ..........................................................16
1.8 Angular Velocity and Angular Acceleration ...................................................18
1.9 Infinitesimal Rotations and the Angular Velocity Vector ...............................19
Chapter 2 Newtonian Mechanics ................................................................................................25
2.1 The First Law of Motion (Law of Inertia) .......................................................25
2.1.1 Inertial Frames of Reference ..............................................................26
2.2 The Second Law of Motion; the Equations of Motion ....................................27
2.2.1 The Concept of Force .........................................................................28
2.3 The Third Law of Motion ................................................................................32
2.3.1 The Concept of Mass ..........................................................................32
2.4 Galilean Transformations and Galilean Invariance ........................................34
2.5 Newton’s Laws of Rotational Motion ..............................................................36
2.6 Work, Energy, and Conservation Laws ...........................................................37
2.6.1 Work and Energy ................................................................................38
2.6.2 Conservative Force and Potential Energy ..........................................39
2.6.3 Conservation of Energy ......................................................................40
2.6.4 Conservation of Momentum ...............................................................42
2.6.5 Conservation of Angular Momentum ................................................42
2.7 Systems of Particles .........................................................................................46
2.7.1 Center of Mass....................................................................................46
2.7.2 Motion of CM .....................................................................................48
2.7.3 Conservation Theorems .....................................................................49
References ..................................................................................................................56
Chapter 3 Integration of Newton’s Equation of Motion ..............................................................57
3.1 Introduction .....................................................................................................57
3.2 Motion Under Constant Force .........................................................................58
vii
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viii Contents
3.3 Force Is a Function of Time ............................................................................63
3.3.1 Impulsive Force and Green’s Function Method .................................66
3.4 Force Is a Function of Velocity .......................................................................67
3.4.1 Motion in a Uniform Magnetic Field .................................................71
3.4.2 Motion in Nearly Uniform Magnetic Field ........................................73
3.5 Force Is a Function of Position ........................................................................74
3.5.1 Bounded and Unbounded Motion ......................................................75
3.5.2 Stable and Unstable Equilibrium .......................................................76
3.5.3 Critical and Neutral Equilibrium .......................................................78
3.6 Time-Varying Mass System (Rocket System) .................................................79
Chapter 4 Lagrangian Formulation of Mechanics: Descriptions of Motion in
Configuration Space ...................................................................................................85
4.1 Generalized Coordinates and Constraints .......................................................85
4.1.1 Generalized Coordinates ....................................................................85
4.1.2 Degrees of Freedom ...........................................................................85
4.1.3 Configuration Space ...........................................................................86
4.1.4 Constraints..........................................................................................86
4.1.4.1 Holonomic and Nonholonomic Constraints .......................86
4.1.4.2 Scleronomic and Rheonomic Constraints ..........................88
4.2 Kinetic Energy in Generalized Coordinates ...................................................88
4.3 Generalized Momentum ..................................................................................90
4.4 Lagrangian Equations of Motion .....................................................................91
4.4.1 Hamilton’s Principle ...........................................................................91
4.4.2 Lagrange’s Equations of Motion from Hamilton’s Principle .............92
4.5 Nonuniqueness of the Lagrangian .................................................................102
4.6 Integrals of Motion and Conservation Laws .................................................104
4.6.1 Cyclic Coordinates and Conservation Theorems .............................104
4.6.2 Symmetries and Conservation Laws ................................................106
4.6.2.1 Homogeneity of Time and Conservation of Energy .........106
4.6.2.2 Spatial Homogeneity and Momentum Conservation ........107
4.6.2.3 Isotropy of Space and Angular Momentum
Conservation ..................................................................108
4.6.2.4 Noether’s Theorem ...........................................................110
4.7 Scale Invariance ............................................................................................111
4.8 Nonconservative Systems and Generalized Potential ...................................112
4.9 Charged Particle in Electromagnetic Field ....................................................112
4.10 Forces of Constraint and Lagrange’s Multipliers ..........................................114
4.11 Lagrangian versus Newtonian Approach to Classical Mechanics ................119
Reference ..................................................................................................................123
Chapter 5 Hamiltonian Formulation of Mechanics: Descriptions of Motion in Phase
Spaces ..................................................................................................................125
5.1 The Hamiltonian of a Dynamic System ........................................................125
5.1.1 Phase Space ......................................................................................126
5.2 Hamilton’s Equations of Motion ...................................................................126
5.2.1 Hamilton’s Equations from Lagrange’s Equations ...........................126
5.2.2 Hamilton’s Equations from Hamilton’s Principle ............................128
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