indd 1 31/1/20 9:56 AM b2530 International Strategic Relations and China’s National Security: World at the Crossroads This page intentionally left blank b2530_FM.indd 6 01-Sep-16 11:03:06 AM Introduction to Classical Mechanics :RUOG6FLHQWLÀF 11750_9789811217432_TP.indd 2 31/1/20 9:56 AM Published by :RUOG6FLHQWL¿F3XEOLVKLQJ&R3WH/WG 7RK7XFN/LQN6LQJDSRUH 86$R৽FH :DUUHQ6WUHHW6XLWH+DFNHQVDFN1- 8.R৽FH 6KHOWRQ6WUHHW&RYHQW*DUGHQ/RQGRQ:&+ +( /LEUDU\RI&RQJUHVV&RQWURO1XPEHU British Library Cataloguing-in-Publication Data $FDWDORJXHUHFRUGIRUWKLVERRNLVDYDLODEOHIURPWKH%ULWLVK/LEUDU\ INTRODUCTION TO CLASSICAL MECHANICS &RS\ULJKWE\:RUOG6FLHQWL¿F3XEOLVKLQJ&R3WH/WG $OOULJKWVUHVHUYHG7KLVERRNRUSDUWVWKHUHRIPD\QRWEHUHSURGXFHGLQDQ\IRUPRUE\DQ\PHDQV HOHFWURQLFRUPHFKDQLFDOLQFOXGLQJSKRWRFRS\LQJUHFRUGLQJRUDQ\LQIRUPDWLRQVWRUDJHDQGUHWULHYDO system now known or to be invented, without written permission from the publisher. )RUSKRWRFRS\LQJRIPDWHULDOLQWKLVYROXPHSOHDVHSD\DFRS\LQJIHHWKURXJKWKH&RS\ULJKW&OHDUDQFH &HQWHU,QF5RVHZRRG'ULYH'DQYHUV0$ 86$,QWKLVFDVHSHUPLVVLRQWRSKRWRFRS\ LVQRWUHTXLUHGIURPWKHSXEOLVKHU ,6%1 ,6%1 SEN )RUDQ\DYDLODEOHVXSSOHPHQWDU\PDWHULDOSOHDVHYLVLW KWWSV ZZZZRUOGVFLHQWL¿FFRPZRUOGVFLERRNVW VXSSO 3ULQWHGLQ6LQJDSRUH Lakshmi - 11750 - Introduction to Classical Mechanics.indd 1 03-02-20 2:16:49 PM January 28, 2020 16:37 BC: 11750 - Introduction to Classical Mechanics Mechintroroot page v For Kay v b2530 International Strategic Relations and China’s National Security: World at the Crossroads This page intentionally left blank b2530_FM.indd 6 01-Sep-16 11:03:06 AM January 28, 2020 16:37 BC: 11750 - Introduction to Classical Mechanics Mechintroroot page vii Preface The author recently published a book entitled Introduction to Electricity and Magnetism [Walecka (2018)]. It is based on an introductory course taught several years ago at Stanford, with over 400 students enrolled. The only requirements were an elementary knowledge of calculus and famil- iarity with vectors and Newton’s laws; the development was otherwise self-contained.
The lectures, although relatively concise, take one from Coulomb’s law to Maxwell’s equations and special relativity in a lucid and logical fashion. The book has an extensive set of accessible problems that enhances and extends the coverage. As an aid to teaching and learning, the solutions to those problems were subsequently published in a separate text [Walecka (2019)]. Although never presented in an actual course, it occurred to the author that it would be fun to compose an equivalent set of lectures, aimed at the very best students, that would serve as a prequel to that Electricity and Magnetism text.
These lectures would assume a good, concurrent, course in calculus and familiarity with basic concepts in physics (say, from a good high-school course); they would otherwise, again, be self-contained. For my own amusement, I did just that. The lectures start with a review of the necessary mathematics and a review of vectors. The idea of an inertial frame is then introduced, and Newton’s laws are stated, with several applications included.
The concepts of energy and angular momentum are introduced, and the analysis is then extended to many-particle systems. The notions of generalized coordinates and Lagrange’s equations are first introduced on the basis that they reproduce Newton’s laws in the chosen examples. After a lecture introducing the calculus of variations, La- grange’s equations are derived from what then serves as the basic principle vii January 28, 2020 16:37 BC: 11750 - Introduction to Classical Mechanics Mechintroroot page viii viii Introduction to Classical Mechanics of classical mechanics —Hamilton’s principle of stationary action. Several more examples are given of lagrangian mechanics.
Hamilton’s equations are similarly first introduced on the basis that they reproduce Lagrange’s equations and Newton’s laws for the chosen examples, and they are then subsequently similarly derived from Hamilton’s principle of stationary action. Several examples are included of hamiltonian mechanics and phase space. A lecture then discusses the transition from the mechanics of discrete particle systems to that of continuous media. Lagrange’s equations for con- tinuous systems are exhibited and then derived from Hamilton’s principle.
The wave motion of a string under tension serves as the paradigm for con- tinuum mechanics, and the analysis extends up through the construction of the energy-momentum tensor and the reflection and radiation of those waves. Irrotational, isentropic fluid flow, where the velocity field is derived from a potential and there is no internal (reversible) heat flow, serves as the final example of lagrangian continuum mechanics. The lagrangian density is constructed. Bernoulli’s equation and the continuity equation for the mass (number) density are then derived from Lagrange’s equations, and they are related back to Newton’s laws for fluid mechanics.
The energy density and energy flux are constructed, and the analysis is then applied to sound waves, where reflection and radiation are again examined. The goal of this text is to provide a clear and concise set of lectures that take one from the introduction and application of Newton’s laws up to Hamilton’s principle and the lagrangian mechanics of continuous systems. This, indeed, provides the point of departure from classical mechanics to modern quantum field theory.1 An extensive set of accessible problems again enhances and extends the coverage. Now readers may feel that this is an overly ambitious goal for a set of introductory lectures on classical mechanics, and it is hard to argue with that.
I did not feel the goals were too ambitious in the case of the Electricity and Magnetism text. It may be that the current lectures are only relevant to a more advanced honors course. Nevertheless, after completing this text and reading it over several times, I am convinced that the whole thing fits together well, and the book serves as a useful text for good students. I do also believe that the current book provides a good introduction to the more advanced mechanics texts, such as [Fetter and Walecka (2003)].
In 1 See, for example, [Walecka (2010)]. January 28, 2020 16:37 BC: 11750 - Introduction to Classical Mechanics Mechintroroot page ix Preface ix addition, I feel the book serves as a good introduction and companion to other standard mechanics texts such as [Kleppner and Kolenkow (2013); Morin (2008); Thornton and Marion (2012); Kibble and Berkshire (2004); Taylor (2004); Landau and Lifshitz (1976); Goldstein et al. I am therefore submitting the present manuscript for publication. It is my hope that students and teachers alike will share some of the pleasure I took in writing this book.
I would like to once again thank my editor, Ms. Lakshmi Narayanan, for her help and support on this project. Williamsburg, Virginia John Dirk Walecka November 4, 2019 Governor’s Distinguished CEBAF Professor of Physics, emeritus College of William and Mary b2530 International Strategic Relations and China’s National Security: World at the Crossroads This page intentionally left blank b2530_FM.indd 6 01-Sep-16 11:03:06 AM January 28, 2020 16:37 BC: 11750 - Introduction to Classical Mechanics Mechintroroot page xi Contents Preface vii 1.4 Length and Direction. Inertial Coordinate Systems 13 4.3 Two-Body Problem.
Energy 21 xi January 28, 2020 16:37 BC: 11750 - Introduction to Classical Mechanics Mechintroroot page xii xii Introduction to Classical Mechanics 6.3 Two-Body Problem .2 Two-Body System .2 Uniform Circular Motion. System of Particles 33 8.4 Rigid-Body Motion .2 Particle on Table Connected to Hanging Mass .3 Bead on a Rotating Hoop .1 Calculus of Variations .1 Another Bead on a Rotating Hoop .2 Cylinder Rolling on Incline Plane. 73 January 28, 2020 16:37 BC: 11750 - Introduction to Classical Mechanics Mechintroroot page xiii Contents xiii 12.1 Oscillations of Particles Connected by Springs .3 Wave Equation for String .1 One-Dimensional Wave Equation .1 Snapshot at Fixed t .2 Disturbance at a Fixed x. Continuum Mechanics of String 91 15.1 Canonical Momentum Density .6 Energy-Momentum Tensor.
Mechanics of Fluids 105 16. 111 January 28, 2020 16:37 BC: 11750 - Introduction to Classical Mechanics Mechintroroot page xiv xiv Introduction to Classical Mechanics 16. Problems 119 Appendix A Numerical Methods 157 Appendix B Significant Names in Classical Mechanics 159 Bibliography 161 Index 163 January 28, 2020 16:37 BC: 11750 - Introduction to Classical Mechanics Mechintroroot page 1 Chapter 1 Introduction The author recently published a book entitled Introduction to Electricity and Magnetism based on a one-quarter, calculus-based course he taught at Stanford some years ago [Walecka (2018)].1 The purpose of the present book, written just for fun, is to design a one-quarter series of lectures on Introduction to Classical Mechanics that could serve as a prequel to the E&M text.2 It is assumed that the reader has taken a good high-school physics course and is familiar with the basics concepts of units, measurements, vectors, etc. It is also assumed that he or she has taken, or is taking, a good course on calculus.1 Physics We again start with some comments on physics.
Physics provides a way of looking at the world. We describe physical phenomena in mathematical terms with the goal of • Correlating phenomena • Predicting new phenomena The description is tested with experiment. Physics is an experimental sci- ence. The payoff is that • The description is either correct or incorrect • The correct results are universal 1 See also [Walecka (2019)].
2 This mechanics course also serves as a nice introduction to the graduate text [Fetter and Walecka (2003)]. 1 January 28, 2020 16:37 BC: 11750 - Introduction to Classical Mechanics Mechintroroot page 2 2 Introduction to Classical Mechanics The goal is to develop a physical law, presented as a mathematical re- lation, usually (but not always) a statement on the instantaneous develop- ment of a system, and then derive, and test, the mathematical consequences of that law. The two towering geniuses of physics are Newton and Einstein — New- ton, who invented calculus to implement his second law, and Einstein, who realized our concepts of space and time depend on how fast one is moving and on any nearby mass.3 We start our discussion with homage to Newton, and give a brief review of the elements of calculus that we will need for our Introduction to Classical Mechanics.2 Calculus Consider the curve described by the function f (x). For every smooth curve, there will be a straight-line tangent to that curve at the point x (Fig.
Now move to a neighboring point x+∆x where ∆x is a very small increment. The function will change to f (x) + ∆f (x).1 Tangent to the curve f (x) at the point x, and increment ∆f (x) in the function when x increases by ∆x. The angle that the tangent to the curve makes with the x-axis (the slope) 3 Maxwell and Schrödinger, with their equations, are not far behind. February 6, 2020 10:53 BC: 11750 - Introduction to Classical Mechanics MechIntroroot page 3 Introduction 3 is given by ∆f (x) tan θ = ; Limit ∆x → 0 (1.1) ∆x where this expression is exact in the limit that the displacement ∆x becomes vanishingly small.
This quantity is called the derivative of f (x) df (x) ∆f (x) ≡ Lim ∆x→0 ; derivative (1.2) dx ∆x Given the value of f (x1 ), one can obain the value f (x) by stepping along the curve (Fig.2 From f (x1 ) to f (x) and area under the curve. This is rewritten as N X (∆f )i f (x) = f (x1 ) + ∆x (1.4) i=1 ∆x The limit N → ∞, which implies ∆x → 0, serves to define the integral February 6, 2020 10:53 BC: 11750 - Introduction to Classical Mechanics MechIntroroot page 4 4 Introduction to Classical Mechanics I(x, x1 ) N X (∆f )i f (x) − f (x1 ) = Lim ∆x→0 ∆x i=1 ∆x ≡ I(x, x1 ) (1.6) ∆x dx and as ∆x → 0 we call it the differential dx, one has Z x df (u) I(x, x1 ) = du ; integral (1.7) x1 du where we have simply re-labeled the dummy integration variable. Stepping past x with a finite interval ∆x gives ∆f (x) I(x + ∆x, x1 ) = I(x, x1 ) + ∆x (1.8) ∆x which is rewritten as I(x + ∆x, x1 ) − I(x, x1 ) ∆f (x) = (1.9) ∆x ∆x The limit ∆x → 0 then gives dI(x, x1 ) df (x) = (1.10) dx dx The derivative of the integral with respect to its upper limit is the integrand evaluated at that upper limit. It is evident from Fig.2 that as the width of each rectangle decreases, and the number of the rectangles increases, one calculates the area under the curve f (x) Z x A(x, x1 ) = f (u)du ; area (1.