Chinmoy Taraphdar - The Classical Mechanics (2007)

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The ClassicalMechanics Chinmoy Taraphdar Asian Books Private Limitedr r The Classical Mechanics Chinmoy Taraphdar Lecturer, Dept. of Physics BankuraChristianCollege Bankura, West Bengal /4Slall?',,,,/U 7JJ'loal,l.t . . l H l l l l . ~7/28, Mahavir Lane, Vardan House, Ansari Road, Daryaganj, N. Delhi-02 Registered and Editorial Office 7/28,Mahavir Lane, Vardan House, Ansari Road,Darya Ganj, Delhi-l 10 002 E-mail:asian@asiabbooksindia.com World Wide Web: http://www.asianbooksindia.com Phones:23287577,23282098,23271887,23259161Fax:911123262021 Sales Offices Bangalore103, Swiss Complex No.33, Race Course Road, Bangalore-560 001 Ph.: 22200438 Fax: 918022256583 E-mail: asianblr@airtelbroadband.in ChennaiPalani Murugan Building No.21, West Cott Road, Royapettah, Chennai-600 014 Ph.:28486927,28486928 E-mail: asianmds@vsnl.net Delhi7/28, Mahavir Lane, Vardan House, Ansari Road, Darya Ganj, Delhi -110 002 Phones:23287577,23282098,23271887,23259161 E-mail:asian@asiabbooksindia.com Guwahati6,G.N.B.Road, Panbazar, Guwahati, Assam-781001 Ph.: 0361-2513020, 2635729 E-mail:asianghyl@sancharnet.in Hyderabad3-5-11Ol/11B I1ndFloor, Opp. Blood Bank, Narayanguda, Hyderabad-500 029 Phones: 24754941, 24750951Fax: 91-40-24751152 Email: hydasian@hd2.vsnl.net.in KolkatalOA, Hospital Street, Calcutta-700 072 Ph.: 22153040 Fax: 91-33-22159899 E-mail: calasian@cal.vsnl.com MumbaiShop No.3 &4, Ground Floor, Shilpin Centre 40, G.D. Ambekar Marg, Sewree Wadala Estate, Wadala, Mumbai-400031 Ph.:91-22-22619322, 22623572 Fax: 91-22-24159899 PuneShop No.5-8, G.F.Shaan Brahma Complex, Budhwar Peth, Pune-02 Publisher Ph.:020-24497208 Fax: 91-20-24497207 E-mail: asianpune@asianbooksindia.com I st Published 2007 ISBN978-81-8412-039-4 All Rights Reserved.No part of this publication may be reproduced, stored in a retrieval system, or transmitted in any formor byanymeans, electronic,mechanical, photocopying, recording and/or otherwise, without the prior written permission of the publisher. Published by Kamal Jagasia for AsianBooks Pvt.Ltd., 7128,Mahavir Lane, VardanHouse, Ansari Road, Darya Ganj, New Delhi-I 10 002. Typeset at : Arpit Printographers, Delhi-32 Printed at .'Gopalji Enterprises, Delhi. Preface This book, organised into ten chapters, is written tocover the syllabus of classical mechanicsforthestudentsof physicsatthegraduateand postgraduatelevel.I hopethattheclear,lucidandcomprehensivecoverageof thisbookwillhelp students togaina thorough grounding of thesubject. Thebeginningpartof thisbookexplainsseveralchaptersonthebasisof Newtonianmechanicsandthenthetextexplainsthegeneralisedco-ordinates andLagrangian mechanicsalong with Hamiltonian mechanicsbasically forthe holonomicsystem.Theconcludingchapterdealswiththecanonical transformations by which thesolution tothe particular problem can be obtained trivially.Someproblemshavebeenworkedouttoaidinunderstandingthe underlying theory attheend of eachchapter. Finally,theexpressionsof acknowledgements.Iamindebtedtomy colleagues,studentswhohavekindlygivenmevaluablecommentsand suggestions.Iacknowledgewithadeepsenseof gratitudemyindebtednessto theauthorswhosestandardworksinthefieldIhavefreelyconsultedtomy benefit. I also acknowledge my indebtedness tomy wife'Anamika', my daughter 'Sreetama' and my son 'Jyotirmoy' for their helpat every stage of the preparation of themanuscript.My specialthanksareduetoallconcerned of 'Asian Books PrivateLimited',especiallytoMs.PurobiBiswas,ProductionManager.Mr. SubhadipKhan,theBranchManagerof Kolkataofficefortheirkindhelpin bringingoutthevolumeinitsadmirableformandbearingwithmeatevery stage withunfailing patience andgood humour. It isferventlyhopedthatthebookwillbeof valuetothestudentsand teachersalike.Commentsandsuggestionsforimprovementstothetext willbe thankfullyacknowledged. ChinmoyTaraphdar "This page is Intentionally Left Blank" Contents Preface Chapter 1.Vector 1.1.Fundamental Concept of Scalar and Vectors,1 1.2.Unit Vectors and General representation of a vector,1 1.3.Multiplication and Division of Vectors by Scalar, 2 1.4.Collinear Vectors, 3 1.5.Linear Dependence or Independence of Vectors, 3 1.6.Addition and Subtraction of two Vectors, 4 1.7.Addition of More Than Two Vectors, 5 1.8.Position Vector and Its Representation in Co-ordinate System, 5 1.9.Condition of Co-planarity of Vectors, 7 1.10.Rotational Invarience of Vector in Reference Frame, 8 1.11.Product of Two Vectors, 8 1.12.Scalar Tripple Product,10 1.13.Vector Tripple Product,11 1.14.Pseudo Vectors and Pseudo Scalars,11 1.15.Vector Derivatives (Ordinary), 13 1.16.Vector Derivatives (Partial) and Vector Operators, 13 1.17.Laplacian and D'Alambertian Operator,18 1.18.Vector Integration,19 1.19.Gauss's Divergence Theorem, 20 1.20.Green's Theorem, 21 1.21.Stoke's Theorem, 22 1.23.Reciprocal Vectors, 24 1.23.Scalar and Vector Field, 24 1.25.Elementary Idea about Vector Space, 24 1.25.Linear Operator in Vector Space, 25 Summary, 26 Worked Out Examples, 29 Exercises, 34 Chapter 2.Linear Motion 2.1.Introduction,37 2.2.Kinematics,37 2.3.Basic Definitions of Required Parameters, 37 2.4.Velocity and Acceleration in Several Co-ordinate System, 40 2.5.Tangential and Normal Component of Velocity and Acceleration,41 2.6.Radialand Transverse Component of Velocity and Acceleration, 42 2.7.Newton's Laws of Motion, 44 2.8.Accelerated Linear Motion, 45 (iii) 1-36 37-64 (vi) 2.9.Graphical Treatment of Linear Motion, 45 2.10.Conservation of Linear Motion, 46 2.11.Time Integral of Force (Impulse), 46 2.12.Work,47 2.13.Power, 48 2.14.Energy:KineticandPotential, 48 2.15.Conservative Force, 49 2.16.Conservation of Energy,49 2.17.Center of Massand Its Motion, 50 2.18.The Two Body Problem, 51 2.19.Application of the Principle of Linear Motion, 52 2.20.Mechanics of Variable Mass, 53 Summary, 55 Worked Out Examples, 58 Exercises, 64 Chapter 3.Rotational Motion: Rigid Body Rotation65-94 3.1.Introduction,65 3.2.Angular velocity and Angular Momentum, 65 3.3.Angular Acceleration, 66 3.4.Moment of Inertia and Torque, 67 3.5.Centrifugal force,70 3.6.RotationalKineticEnergy,71 3.7.Angular Momentum forRigid Body Rotation,72 3.8.KineticEnergy forRigidBody Rotation,74 3.9.Axes theorem for Moment of Inertia, 75 3.10.Calculation of Moment of Inertia in different cases, 76 3.11.Momental Ellipsoid or Ellipsoid of Inertia,83 3.13.Moment andproduct of Inertia andEllipsoid of inertia of some, symmetrical bodies,84 3.12.Moment of Inertia Tensor,87 3.14.Routh's Rule,87 3.15.Euler's Angles, 88 Summary,89 Worked Out Examples, 91 Exercises, 93 Chapter 4.Reference Frame95-111 4.1.Introduction,95 4.2.Non Inertial Frame and Pseudo Force, 95 4.3.Effect of rotation of earth onacceleration due togravity,99 4.4.Effect of Coriolis Force on a particle moving on the surface of earth,101 4.5.Effect of Coriolis force on a particle falling freelyunder gravity,103 4.6.Principle of Foucault's Pendulum,104 4.7.Flow of River on Earth Surface,106 Summary, 106 Worked out Examples,107 Exercises,110 (vii) Chapter 5.Central Force112-124 5.1.Introduction,112 5.2.Definition and Characteristics of Central force,112 5.3.Conservation of Angular Momentum under Central Force,113 5.4.Conservation of energy under central force,113 5.5.Equation of motion under attractive central force,115 5.6.Application of central forcetheorytogravitation Deduction of Keplar's law,116 5.7.Energy conservation forplanetory motion,118 5.8.Stability of Orbits,120 Summary, 120 Worked out Examples,121 Exercises,124 Chapter 6.Theory of Collision125-137 6.1.Introduction,125 6.2.Characteristicsof Collision,125 6.3.Lab Frame and Center of Mass Frame,126 6.4.Direct or Linear Collision,127 6.5.Characteristic of Direct Collision,129 6.6.Maximum Energytransfer due tohead on elastic collision,130 6.7.ObliqueCollision,131 Summary, 132 Worked Out Examples,133 Exercises,137 Chapter 7.Conservation Principle and Constrained Motion138-154 7.1.Characteristicsof Conservation Principle,138 7.2.Mechanics of a single particle andsystem of particles,139 7.3.Conservation of linear momentum,140 7.4.Conservation of Angular Momentum,141 7.5.Conservation of Energy,142 7.6.ConstrainedMotion,145 7.7.Generalised Co-ordinates and other Generalised Parameters,146 7.8.Limitation of Newton's Law,151 Summary, 151 Worked out examples,153 Exercises,154 Chapter 8.Variational Principle and Lagrangian Mechanics155-193 8.1.Introduction,155 8.2.Forces of Constraint,155 8.3.VirtualDisplacement,156 8.4.Principle of Virtual Work,156 8.5.D' Alembert's Principle, 157 8.6.Lagrange's equations for a holonomic System,158 8.7.Lagrange's equation fora conservative,non-holonomicsystem,160 8.8.IntroductiontoCalculusof variations,161 8.9.VariationalTechnique formanyindependent variables: Euler-Lagrange'sdifferentialequation,165 ( viii) 8.10.Hamilton's Variational Principle,166 8.11.Derivation of Hamilton's principle from Lagrange's equation,167 8.12.Derivation of Lagrange's equations from Hamilton's principle,168 8.13.Derivation of Lagrange's equation from D' Alambert's principle,169 8.14.Derivation of Hamilton's Principle from D' Alambert's Principle,171 8.15.Cyclic or Ignorable Co-ordinates,172 8.16.Conservation Theorems,172 8.17.Gauge Function for Lagrangian,175 8.18.Invarience of Lagrange's equations under Generalised Co-ordinate, transformations,177 8.19.Concept of Symmetry: Homogeneity and Isotropy,178 8.20.Invarience of Lagrange's equation under Galilean Transformation,179 8.21.Application's of Lagrange's equation of motion in several mechanical systems,180 Summary, 186 Worked Out Examples,187 Exercises,193 Chapter 9.Hamiltonian Formulation in Mechanics194-209 9.1.Introduction,194 9.2.Hamiltonian of the System,194 9.3.Concept of Phase Space,195 9.5.Hamilton's Canonical Equations in different Co-ordinate System,197 9.6.Hamilton's Canonical equations from Hamilton's Intergral Principle,199 9.7.Physical Significance of Hamiltonian of the System, 201 9.8.Advantage of Hamiltonian Approach,201 9.9.Principle of Least Action, 201 9.10.Difference betwee