MECHANICS Classical Mechanics

Classical MechanicsSecond rbitFinalspacecraftorbitOuterplanetorbitTai L. Chow

Classical MechanicsSecond Edition

Classical MechanicsSecond EditionTai L. ChowBoca Raton London New YorkCRC Press is an imprint of theTaylor & Francis Group, an informa business

CRC PressTaylor & Francis Group6000 Broken Sound Parkway NW, Suite 300Boca Raton, FL 33487-2742 2013 by Taylor & Francis Group, LLCCRC Press is an imprint of Taylor & Francis Group, an Informa businessNo claim to original U.S. Government worksVersion Date: 20130227International Standard Book Number-13: 978-1-4665-7000-9 (eBook - PDF)This book contains information obtained from authentic and highly regarded sources. Reasonable efforts have beenmade to publish reliable data and information, but the author and publisher cannot assume responsibility for the validity of all materials or the consequences of their use. The authors and publishers have attempted to trace the copyrightholders of all material reproduced in this publication and apologize to copyright holders if permission to publish in thisform has not been obtained. If any copyright material has not been acknowledged please write and let us know so we mayrectify in any future reprint.Except as permitted under U.S. Copyright Law, no part of this book may be reprinted, reproduced, transmitted, or utilized in any form by any electronic, mechanical, or other means, now known or hereafter invented, including photocopying, microfilming, and recording, or in any information storage or retrieval system, without written permission from thepublishers.For permission to photocopy or use material electronically from this work, please access www.copyright.com (http://www.copyright.com/) or contact the Copyright Clearance Center, Inc. (CCC), 222 Rosewood Drive, Danvers, MA 01923,978-750-8400. CCC is a not-for-profit organization that provides licenses and registration for a variety of users. Fororganizations that have been granted a photocopy license by the CCC, a separate system of payment has been arranged.Trademark Notice: Product or corporate names may be trademarks or registered trademarks, and are used only foridentification and explanation without intent to infringe.Visit the Taylor & Francis Web site athttp://www.taylorandfrancis.comand the CRC Press Web site athttp://www.crcpress.com

ToRobert YoungmeandDavid Lori

ContentsPreface. xvAuthor.xviiChapter 1Kinematics: Describing the Motion.11.11.21.31.41.51.61.71.81.9Chapter 2Introduction. 1Space, Time, and Coordinate Systems.1Change of Coordinate System (Transformation of Components of aVector). 3Displacement Vector.8Speed and Velocity. 8Acceleration. 101.6.1 Tangential and Normal Acceleration. 11Velocity and Acceleration in Polar Coordinates. 141.7.1 Plane Polar Coordinates (r, θ). 141.7.2 Cylindrical Coordinates (ρ, ϕ, z). 151.7.3 Spherical Coordinates (r , θ, ϕ). 16Angular Velocity and Angular Acceleration. 18Infinitesimal Rotations and the Angular Velocity Vector. 19Newtonian Mechanics.252.1The First Law of Motion (Law of Inertia).252.1.1 Inertial Frames of Reference.262.2 The Second Law of Motion; the Equations of Motion. 272.2.1 The Concept of Force.282.3 The Third Law of Motion. 322.3.1 The Concept of Mass. 322.4 Galilean Transformations and Galilean Invariance.342.5 Newton’s Laws of Rotational Motion. 362.6 Work, Energy, and Conservation Laws. 372.6.1 Work and Energy. 382.6.2 Conservative Force and Potential Energy. 392.6.3 Conservation of Energy.402.6.4 Conservation of Momentum. 422.6.5 Conservation of Angular Momentum. 422.7 Systems of Particles.462.7.1 Center of Mass.462.7.2 Motion of CM. 482.7.3 Conservation Theorems. 49References. 56Chapter 3Integration of Newton’s Equation of Motion. 573.13.2Introduction. 57Motion Under Constant Force. 58vii 2010 Taylor & Francis Group, LLC

viiiContents3.33.43.53.6Chapter 4Force Is a Function of Time. 633.3.1 Impulsive Force and Green’s Function Method.66Force Is a Function of Velocity. 673.4.1 Motion in a Uniform Magnetic Field. 713.4.2 Motion in Nearly Uniform Magnetic Field. 73Force Is a Function of Position. 743.5.1 Bounded and Unbounded Motion. 753.5.2 Stable and Unstable Equilibrium. 763.5.3 Critical and Neutral Equilibrium. 78Time-Varying Mass System (Rocket System). 79Lagrangian Formulation of Mechanics: Descriptions of Motion inConfiguration Space. 854.1Generalized Coordinates and Constraints. 854.1.1 Generalized Coordinates. 854.1.2 Degrees of Freedom. 854.1.3 Configuration Space. 864.1.4 Constraints. 864.1.4.1 Holonomic and Nonholonomic Constraints. 864.1.4.2 Scleronomic and Rheonomic Constraints. 884.2 Kinetic Energy in Generalized Coordinates. 884.3 Generalized Momentum.904.4 Lagrangian Equations of Motion. 914.4.1 Hamilton’s Principle. 914.4.2 Lagrange’s Equations of Motion from Hamilton’s Principle.924.5 Nonuniqueness of the Lagrangian. 1024.6 Integrals of Motion and Conservation Laws. 1044.6.1 Cyclic Coordinates and Conservation Theorems. 1044.6.2 Symmetries and Conservation Laws. 1064.6.2.1 Homogeneity of Time and Conservation of Energy. 1064.6.2.2 Spatial Homogeneity and Momentum Conservation. 1074.6.2.3 Isotropy of Space and Angular MomentumConservation. 1084.6.2.4 Noether’s Theorem. 1104.7 Scale Invariance. 1114.8 Nonconservative Systems and Generalized Potential. 1124.9 Charged Particle in Electromagnetic Field. 1124.10 Forces of Constraint and Lagrange’s Multipliers. 1144.11 Lagrangian versus Newtonian Approach to Classical Mechanics. 119Reference. 123Chapter 5Hamiltonian Formulation of Mechanics: Descriptions of Motion in PhaseSpaces. 1255.15.2The Hamiltonian of a Dynamic System. 1255.1.1 Phase Space. 126Hamilton’s Equations of Motion. 1265.2.1 Hamilton’s Equations from Lagrange’s Equations. 1265.2.2 Hamilton’s Equations from Hamilton’s Principle. 128 2010 Taylor & Francis Group, LLC

ixContents5.3Integrals of Motion and Conservation Theorems. 1325.3.1 Energy Integrals. 1325.3.2 Cyclic Coordinates and Integrals of Motion. 1325.3.3 Conservation Theorems of Momentum and AngularMomentum.1335.4 Canonical Transformations. 1355.5 Poisson Brackets. 1405.5.1 Fundamental Properties of Poisson Brackets. 1415.5.2 Fundamental Poisson Brackets. 1415.5.3 Poisson Brackets and Integrals of Motion. 1415.5.4 Equations of Motion in Poisson Bracket Form. 1445.5.5 Canonical Invariance of Poisson Brackets. 1445.6 Poisson Brackets and Quantum Mechanics. 1455.7 Phase Space and Liouville’s Theorem. 1475.8 Time Reversal in Mechanics (Optional). 1505.9 Passage from Hamiltonian to Lagrangian. 151References. 154Chapter 6Motion Under a Central Force. 1556.16.26.36.4Two-Body Problem and Reduced Mass. 155General Properties of Central Force Motion. 157Effective Potential and Classification of Orbits. 159General Solutions of Central Force Problem. 1636.4.1 Energy Method. 1636.4.2 Lagrangian Analysis. 1646.5 Inverse Square Law of Force. 1676.6 Kepler’s Three Laws of Planetary Motion. 1726.7 Applications of Central Force Motion. 1746.7.1 Satellites and Spacecraft. 1746.7.2 Communication Satellites. 1786.7.3 Flyby Missions to Outer Planets. 1796.8 Newton’s Law of Gravity from Kepler’s Laws. 1826.9 Stability of Circular Orbits (Optional). 1836.10 Apsides and Advance of Perihelion (Optional). 1886.10.1 Advance of Perihelion and Inverse-Square Force. 1896.10.2 Method of Perturbation Expansion. 1906.11 Laplace–Runge–Lenz Vector and the Kepler Orbit (Optional). 192References. 198Chapter 7Harmonic Oscillator. 1997.17.27.3Simple Harmonic Oscillator. 1997.1.1 Motion of Mass m on the End of a Spring. 1997.1.2 The Bob of Simple Pendulum Swinging through a Small Arc.2007.1.3 Solution of Equation of Motion of SHM. 2017.1.4 Kinetic, Potential, Total, and Average Energies of HarmonicOscillator. 203Adiabatic Invariants and Quantum Condition.206Damped Harmonic Oscillator.209 2010 Taylor & Francis Group, LLC

xContents7.47.57.6Phase Diagram for Damped Oscillator. 218Relaxation Time Phenomena. 220Forced Oscillations without Damping. 2207.6.1 Periodic Driving Force. 2217.6.2 Arbitrary Driving Forces. 2237.7 Forced Oscillations with Damping. 2257.7.1 Resonance. 2277.7.2 Power Absorption. 2317.8 Oscillator Under Arbitrary Periodic Force. 2357.8.1 Fourier’s Series Solution. 2367.9 Vibration Isolation. 2397.10 Parametric Excitation. 241Chapter 8Coupled Oscillations and Normal Coordinates. 2498.18.28.38.4Chapter 9Coupled Pendulum. 2498.1.1 Normal Coordinates. 251Coupled Oscillators and Normal Modes: General Analytic Approach. 2548.2.1 The Equation of Motion of a Coupled System. 2548.2.2 Normal Modes of Oscillation. 2558.2.3 Orthogonality of Eigenvectors. 2578.2.4 Normal Coordinates. 259Forced Oscillations of Coupled Oscillators.264Coupled Electric Circuits.266Nonlinear Oscillations. 2739.19.29.3Qualitative Analysis: Energy and Phase Diagrams. 274Elliptical Integrals and Nonlinear Oscillations.280Fourier Series Expansions. 2839.3.1 Symmetrical Potential: V(x) V( x).2849.3.2 Asymmetrical Potential: V( x) V(x). 2879.4 The Method of Perturbation. 2889.4.1 Bogoliuboff–Kryloff Procedure and Removal of Secular Terms.2929.5 Ritz Method. 2959.6 Method of Successive Approximation. 2979.7 Multiple Solutions and Jumps. 2999.8 Chaotic Oscillations. 3019.8.1 Some Helpful Tools for an Understanding of Chaos. 3019.8.2 Conditions for Chaos.3069.8.3 Routes to Chaos.3079.8.4 Lyapunov Exponentials.308References. 312Chapter 10 Collisions and Scatterings. 31310.1 Direct Impact of Two Particles. 31310.2 Scattering Cross Sections and Rutherford Scattering. 31810.2.1 Scattering Cross Sections. 31910.2.2 Rutherford’s α-Particle Scattering Experiment. 32010.2.3 Cross Section Is Lorentz Invariant. 324 2010 Taylor & Francis Group, LLC

xiContents10.3 Laboratory and Center-of-Mass Frames of Reference. 32410.4 Nuclear Sizes. 32810.5 Small-Angle Scattering (Optional). 329References. 336Chapter 11 Motion in Non-Inertial Systems. 33711.1 Accelerated Translational Coordinate System. 33711.2 Dynamics in Rotating Coordinate System. 34111.2.1 Centrifugal Force. 34511.2.2 The Coriolis Force. 34911.2.2.1 Trade Winds and Circulation of Ocean Currents. 35111.2.2.2 Weather Systems. 35211.2.2.3 Hurricanes. 35411.2.2.4 Bathtub Vortex and Earth Rotation. 35411.3 Motion of Particle Near the Surface of the Earth. 35511.4 Foucault Pendulum. 36111.5 Larmor’s Theorem.36411.6 Classical Zeeman Effect. 36511.7 Principle of Equivalence. 36811.7.1 Principle of Equivalence and Gravitational Red Shift. 369Chapter 12 Motion of Rigid Bodies. 37712.112.212.312.412.5Independent Coordinates of Rigid Body. 378Eulerian Angles. 379Rate of Change of Vector. 382Rotational Kinetic Energy and Angular Momentum. 384Inertia Tensor. 39412.5.1 Diagonalization of a Symmetric Tensor. 39612.5.2 Moments and Products of Inertia. 39712.5.3 Parallel-Axis Theorem. 39812.5.4 Moments of Inertia about an Arbitrary Axis. 40112.5.5 Principal Axes of Inertia.40312.6 Euler’s Equations of Motion.40712.7 Motion of a Torque-Free Symmetrical Top.40912.8 Motion of Heavy Symmetrical Top with One Point Fixed. 41412.8.1 Precession without Nutation. 41712.8.2 Precession with Nutation. 41912.9 Stability of Rotational Motion. 420References. 425Chapter 13 Theory of Special Relativity. 42713.1 Historical Origin of Special Theory of Relativity. 42713.2 Michelson–Morley Experiment. 43013.3 Postulates of Special Theory of Relativity. 43313.3.1 Time Is Not Absolute. 43413.4 Lorentz Transformations. 43413.4.1 Relativity of Simultaneity, Causality. 43713.4.2 Time Dilation, Relativity of Co-Locality. 438 2010 Taylor & Francis Group, LLC