Foundations of Classical Mechanics Classical mechanics is an old subject. It is being taught differently in modern times using numerical examples and exploiting the power of computers that are readily available to modern students. The demand of the twenty-first century is challenging. The previous century saw revolutionary developments, and the present seems poised for further exciting insights and illuminations. Students need to quickly grasp the foundations and move on to an advanced level of maturity in science and engineering. Freshman students who have just finished high school need a short and rapid, but also thorough and comprehensive, exposure to major advances, not only since Copernicus, Kepler, Galileo, and Newton, but also since Maxwell, Einstein, and Schrodinger. Through a careful selection of topics, this book endeavors to induct freshman students into this excitement. This text does not merely teach the laws of physics; it attempts to show how they were unraveled. Thus, it brings out how empirical data led to Kepler’s laws, to Galileo’s law of inertia, how Newton’s insights led to the principle of causality and determinism. It illustrates how symmetry considerations lead to conservation laws, and further, how the laws of nature can be extracted from these connections. The intimacy between mathematics and physics is revealed throughout the book, with an emphasis on beauty, elegance, and rigor. The role of mathematics in the study of nature is further highlighted in the discussions on fractals and Madelbrot sets. In its formal structure the text discusses essential topics of classical mechanics such as the laws of Newtonian mechanics, conservation laws, symmetry principles, Euler–Lagrange equation, wave motion, superposition principle, and Fourier analysis. It covers a substantive introduction to fluid mechanics, electrodynamics, special theory of relativity, and to the general theory of relativity. A chapter on chaos explains the concept of exploring laws of nature using Fibonacci sequence, Lyapunov exponent, and fractal dimensions. A large number of solved and unsolved exercises are included in the book for a better understanding of concepts. P. C. Deshmukh is a Professor in the Department of Physics at the Indian Institute of Technology Tirupati, India, and concurrently an Adjunct Faculty at the Indian Institute of Technology Madras, India. He has taught courses on classical mechanics, atomic and molecular physics, quantum mechanics, solid state theory, foundations of theoretical physics, and theory of atomic collisions at undergraduate and graduate levels. Deshmukh’s current research includes studies in ultrafast photoabsorption processes in free/confined atoms, molecules, and ions using relativistic quantum many-electron formalism. He also works on the applications of the Lambert W function in pure and applied physics. Foundations of Classical Mechanics P. C. Deshmukh University Printing House, Cambridge CB2 8BS, United Kingdom One Liberty Plaza, 20th Floor, New York, NY 10006, USA 477 Williamstown Road, Port Melbourne, vic 3207, Australia 314 to 321, 3rd Floor, Plot No.3, Splendor Forum, Jasola District Centre, New Delhi 110025, India 79 Anson Road, #06–04/06, Singapore 079906 Cambridge University Press is part of the University of Cambridge. It furthers the University’s mission by disseminating knowledge in the pursuit of education, learning and research at the highest international levels of excellence. www.cambridge.org Information on this title: www.cambridge.org/9781108480567 P. C. Deshmukh 2019 This publication is in copyright. Subject to statutory exception and to the provisions of relevant collective licensing agreements, no reproduction of any part may take place without the written permission of Cambridge University Press. First published 2019 Printed in India A catalogue record for this publication is available from the British Library ISBN 978-1-108-48056-7 Hardback ISBN 978-1-108-72775-4 Paperback Cambridge University Press has no responsibility for the persistence or accuracy of URLs for external or third-party internet websites referred to in this publication, and does not guarantee that any content on such websites is, or will remain, accurate or appropriate. To my teachers, with much gratitude and to my students, with best wishes Contents List of Figures xi List of Tables xxiii Foreword xxv Preface xxvii Introduction 1 1. Laws of Mechanics and Symmetry Principles 9 1.1 The Fundamental Question in Mechanics 9 1.2 The Three Laws of Newtonian Mechanics 11 1.3 Newton’s Third Law: an Alternative Viewpoint 19 1.4 Connections between Symmetry and Conservation Laws 20 2. Mathematical Preliminaries 31 2.1 M athematics, the Natural Ally of Physics 31 2.2 Equation of Motion in Cylindrical and Spherical Polar Coordinate Systems 40 2.3 Curvilinear Coordinate Systems 50 2.4 Contravariant and Covariant Tensors 57 Appendix: Some important aspects of Matrix Algebra 61 3. Real Effects of Pseudo-forces: Description of Motion in Accelerated Frame of Reference 78 3.1 Galileo–Newton Law of Inertia—Again 78 3.2 Motion of an Object in a Linearly Accelerated Frame of Reference 81 3.3 Motion of an Object in a Rotating Frame of Reference 84 3.4 Real Effects That Seek Pseudo-causes 90 4. Small Oscillations and Wave Motion 111 4.1 Bounded Motion in One-dimension, Small Oscillations 111 4.2 Superposition of Simple Harmonic Motions 127 4.3 Wave Motion 133 4.4 Fourier Series, Fourier Transforms, and Superposition of Waves 137 viii Contents 5. Damped and Driven Oscillations; Resonances 156 5.1 Dissipative Systems 156 5.2 Damped Oscillators 157 5.3 Forced Oscillations 168 5.4 Resonances, and Quality Factor 172 6. The Variational Principle 183 6.1 The Variational Principle and Euler–Lagrange’s Equation of Motion 183 6.2 The Brachistochrone 193 6.3 Supremacy of the Lagrangian Formulation of Mechanics 199 6.4 Hamilton’s Equations and Analysis of the S.H.O. Using Variational Principle 208 Appendix: Time Period for Large Oscillations 212 7. Angular Momentum and Rigid Body Dynamics 225 7.1 The Angular Momentum 225 7.2 The Moment of Inertia Tensor 233 7.3 Torque on a Rigid Body: Euler Equations 242 7.4 The Euler Angles and the Gyroscope Motion 250 8. The Gravitational Interaction in Newtonian Mechanics 267 8.1 Conserved Quantities in the Kepler–Newton Planetary Model 267 8.2 Dynamical Symmetry in the Kepler–Newton Gravitational Interaction 275 8.3 Kepler–Newton Analysis of the Planetary Trajectories about the Sun 279 8.4 More on Motion in the Solar System, and in the Galaxy 284 Appendix: The Repulsive ‘One-Over-Distance’ Potential 290 9. Complex Behavior of Simple System 306 9.1 Learning from Numbers 306 9.2 Chaos in Dynamical Systems 312 9.3 Fractals 322 9.4 Characterization of Chaos 336 10. Gradient Operator, Methods of Fluid Mechanics, and Electrodynamics 347 10.1 T he Scalar Field, Directional Derivative, and Gradient 347 10.2 T he ‘Ideal’ Fluid, and the Current Density Vector Field 355 10.3 G auss’ Divergence Theorem, and the Conservation of Flux 362 10.4 V orticity, and the Kelvin–Stokes’ Theorem 372 11. Rudiments of Fluid Mechanics 384 11.1 T he Euler’s Equation of Motion for Fluid Flow 384 11.2 Fluids at Rest, ‘Hydrostatics’ 392 11.3 S treamlined Flow, Bernoulli’s Equation, and Conservation of Energy 401 11.4 N avier–Stokes Equation(s): Governing Equations of Fluid Motion 405