Table Of Content

Lagrangian Mechanics To our parents; to Nicole and Younnik, to Mai; to Yemma Nounou, Shehrazed, Rayane, Redwan, Elyas Series Editor Noël Challamel Lagrangian Mechanics An Advanced Analytical Approach Anh Le van Rabah Bouzidi First published 2019 in Great Britain and the United States by ISTE Ltd and John Wiley & Sons, Inc. Apart from any fair dealing for the purposes of research or private study, or criticism or review, as permitted under the Copyright, Designs and Patents Act 1988, this publication may only be reproduced, stored or transmitted, in any form or by any means, with the prior permission in writing of the publishers, or in the case of reprographic reproduction in accordance with the terms and licenses issued by the CLA. Enquiries concerning reproduction outside these terms should be sent to the publishers at the undermentioned address: ISTE Ltd John Wiley & Sons, Inc. 27-37 St George’s Road 111 River Street London SW19 4EU Hoboken, NJ 07030 UK USA www.iste.co.uk www.wiley.com © ISTE Ltd 2019 The rights of Anh Le van and Rabah Bouzidi to be identified as the authors of this work have been asserted by them in accordance with the Copyright, Designs and Patents Act 1988. Library of Congress Control Number: 2019937362 British Library Cataloguing-in-Publication Data A CIP record for this book is available from the British Library ISBN 978-1-78630-436-0 Contents Preface xi 1 Kinematics 1 1.1 Observer–Referenceframe . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 1.2 Time . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 1.2.1 Datepostulate . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 1.2.2 Datechangepostulate . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 1.3 Space . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 1.3.1 Physicalspace . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 1.3.2 Mathematicalspace . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 1.3.3 Positionpostulate . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 1.3.4 TypicaloperationsonthemathematicalspaceE . . . . . . . . . . . . . . . 6 1.3.5 Positionchangepostulate . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 1.3.6 ThecommonreferenceframeR0 . . . . . . . . . . . . . . . . . . . . . . . 9 1.3.7 Coordinatesystemofareferenceframe . . . . . . . . . . . . . . . . . . . . 12 1.3.8 Fixedpointandfixedvectorinareferenceframe . . . . . . . . . . . . . . . 14 1.4 Derivativeofavectorwithrespecttoareferenceframe. . . . . . . . . . . . . . . 15 1.5 Velocityofaparticle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17 1.6 Angularvelocity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17 1.7 Referenceframedefinedbyarigidbody:Rigidbodydefinedbyareference frame . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19 1.8 Pointattachedtoarigidbody:Vectorattachedtoarigidbody . . . . . . . . . . . 19 1.9 Velocitiesinarigidbody. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20 1.10Velocitiesinamechanicalsystem . . . . . . . . . . . . . . . . . . . . . . . . . . 22 1.11Acceleration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24 1.11.1 Accelerationofaparticle . . . . . . . . . . . . . . . . . . . . . . . . . . . 24 1.11.2 Accelerationsinamechanicalsystem . . . . . . . . . . . . . . . . . . . . . 24 1.12Compositionofvelocitiesandaccelerations . . . . . . . . . . . . . . . . . . . . . 24 1.12.1 Compositionofvelocities . . . . . . . . . . . . . . . . . . . . . . . . . . . 24 1.12.2 Compositionofaccelerations . . . . . . . . . . . . . . . . . . . . . . . . . 25 1.13Angularmomentum:Dynamicmoment . . . . . . . . . . . . . . . . . . . . . . . 25 2 ParameterizationandParameterizedKinematics 27 2.1 Positionparameters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27 2.1.1 Positionparametersofaparticle . . . . . . . . . . . . . . . . . . . . . . . . 27 2.1.2 Positionparametersforarigidbody . . . . . . . . . . . . . . . . . . . . . . 28 2.1.3 Positionparametersforasystemofrigidbodies . . . . . . . . . . . . . . . 32 vi LagrangianMechanics 2.2 Mechanicaljoints . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 2.3 Constraintequations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 2.4 Parameterization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 2.5 Dependenceoftherotationtensorofthereferenceframeontheretained parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39 2.6 Velocityofaparticle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41 2.7 Angularvelocity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44 2.8 Velocitiesinarigidbody. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45 2.9 Velocitiesinamechanicalsystem . . . . . . . . . . . . . . . . . . . . . . . . . . 47 2.10Parameterizedvelocityofaparticle . . . . . . . . . . . . . . . . . . . . . . . . . 48 2.10.1 Definition. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48 2.10.2 Practicalcalculationoftheparameterizedvelocity . . . . . . . . . . . . . . 49 2.11Parameterizedvelocitiesinarigidbody . . . . . . . . . . . . . . . . . . . . . . . 50 2.12Parameterizedvelocitiesinamechanicalsystem . . . . . . . . . . . . . . . . . . 51 2.13Lagrange’skinematicformula . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52 2.14Parameterizedkineticenergy . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54 3 Efforts 57 3.1 Forces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57 3.2 Torque . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59 3.3 Efforts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60 3.4 Externalandinternalefforts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61 3.4.1 Externaleffort . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61 3.4.2 Internaleffort. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62 3.5 Giveneffortsandconstraintefforts . . . . . . . . . . . . . . . . . . . . . . . . . 62 3.6 Momentfield . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64 4 VirtualKinematics 67 4.1 Virtualderivativeofavectorwithrespecttoareferenceframe . . . . . . . . . . . 67 4.2 Virtualvelocityofaparticle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70 4.3 Virtualangularvelocity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75 4.4 Virtualvelocitiesinarigidbody . . . . . . . . . . . . . . . . . . . . . . . . . . . 81 4.4.1 Thevirtualvelocityfield(VVF)associatedwithaparameterization . . . . . 81 4.4.2 Virtualvelocityfield(VVF)inarigidbody . . . . . . . . . . . . . . . . . . 82 4.5 Virtualvelocitiesinasystem . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83 4.5.1 VVFassociatedwithaparameterization. . . . . . . . . . . . . . . . . . . . 83 4.5.2 VVFoneachrigidbodyofasystem. . . . . . . . . . . . . . . . . . . . . . 84 4.5.3 Virtualvelocityofthecenterofmass . . . . . . . . . . . . . . . . . . . . . 84 4.6 Compositionofvirtualvelocities . . . . . . . . . . . . . . . . . . . . . . . . . . 85 4.6.1 Compositionofvirtualvelocitiesofaparticle . . . . . . . . . . . . . . . . . 85 4.6.2 Compositionofvirtualangularvelocities . . . . . . . . . . . . . . . . . . . 86 4.6.3 CompositionofVVFsinrigidbodies . . . . . . . . . . . . . . . . . . . . . 87 4.7 Methodofcalculatingthevirtualvelocityatapoint . . . . . . . . . . . . . . . . 88 5 VirtualPowers 91 5.1 Principleofvirtualpowers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91 5.2 VPofeffortsinternaltoeachrigidbody . . . . . . . . . . . . . . . . . . . . . . . 92 5.3 VPofefforts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92 5.4 VPofeffortsexertedonarigidbody . . . . . . . . . . . . . . . . . . . . . . . . 94 5.4.1 Generalexpression . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94 5.4.2 VPofzeromomentfieldeffortsexerteduponarigidbody . . . . . . . . . . 94 5.4.3 DependenceoftheVPofeffortsonthereferenceframe . . . . . . . . . . . 94 Contents vii 5.5 VPofeffortsexertedonasystemofrigidbodies . . . . . . . . . . . . . . . . . . 95 5.5.1 Generalexpression . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95 5.5.2 DependenceoftheVPoftheeffortsonthereferenceframe . . . . . . . . . 96 5.5.3 VPofzeromomentfieldeffortsexertedonasystemofrigidbodies . . . . . 96 5.5.4 VPofinter-effortsbetweentherigidbodiesofasystem . . . . . . . . . . . 96 5.5.5 Thespecificcaseoftheinter-effortsbetweentworigidbodies . . . . . . . . 97 5.6 SummaryofthecaseswheretheVVandVPareindependentofthereference frame . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98 5.7 VPofeffortsexpressedasalinearformoftheq˙∗ . . . . . . . . . . . . . . . . . . 99 i 5.8 Potential . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101 5.8.1 Definition. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101 5.8.2 Examplesofpotential . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101 5.9 VPofthequantitiesofacceleration . . . . . . . . . . . . . . . . . . . . . . . . . 108 6 Lagrange’sEquations 111 6.1 ChoiceofthecommonreferenceframeR0 . . . . . . . . . . . . . . . . . . . . . 111 6.2 Lagrange’sequations. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112 6.3 Reviewandtheneedtomodeljoints. . . . . . . . . . . . . . . . . . . . . . . . . 115 6.4 Existenceanduniquenessofthesolution . . . . . . . . . . . . . . . . . . . . . . 116 6.5 Equationsofmotion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 118 6.6 Example1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 118 6.7 Example2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 120 6.7.1 Reducedparameterization . . . . . . . . . . . . . . . . . . . . . . . . . . . 120 6.7.2 Totalparameterization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121 6.7.3 Comparingthetwoparameterizations . . . . . . . . . . . . . . . . . . . . . 122 6.8 Example3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123 6.8.1 Independentparameterization . . . . . . . . . . . . . . . . . . . . . . . . . 124 6.8.2 Totalparameterization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 126 6.8.3 Comparingthetwoparameterizations . . . . . . . . . . . . . . . . . . . . . 127 6.9 Workinginanon-Galileanreferenceframe . . . . . . . . . . . . . . . . . . . . . 127 7 PerfectJoints 131 7.1 VFscompatiblewithamechanicaljoint . . . . . . . . . . . . . . . . . . . . . . . 132 7.1.1 Definition. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 132 7.1.2 GeneralizingthedefinitionofacompatibleVVF . . . . . . . . . . . . . . . 134 7.1.3 Example1forVVFscompatiblewithamechanicaljoint . . . . . . . . . . . 135 7.1.4 Example2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 136 7.1.5 Example3:particlemovingalongahooprotatingaroundafixedaxis . . . . 137 7.2 InvarianceofthecompatibleVVFswithrespecttothechoiceoftheprimitive parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 139 7.2.1 Context . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 139 7.2.2 Relationshipsbetweentherealquantitiesresultingfromthetwo parameterizations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 140 7.2.3 Relationshipsbetweenthevirtualquantitiesresultingfromthetwo parameterizations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 141 7.2.4 IdentitybetweentheVVFsassociatedwiththetwoparameterizationsand compatiblewithamechanicaljoint . . . . . . . . . . . . . . . . . . . . . . 143 7.2.5 Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 145 viii LagrangianMechanics 7.3 InvarianceofthecompatibleVVFswithrespecttothechoiceoftheretained parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 148 7.3.1 Context . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 148 7.3.2 Relationshipsbetweentherealquantitiesresultingfromthetwo parameterizations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 150 7.3.3 Relationshipsbetweenthevirtualquantitiesresultingfromthetwo parameterizations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 151 7.3.4 IdentitybetweentheVVFsassociatedwiththetwoparamaterizationsand compatiblewithamechanicaljoint . . . . . . . . . . . . . . . . . . . . . . 153 7.3.5 Example1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 155 7.3.6 Example2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 156 7.3.7 Example3:particlemovingalongahooprotatingaroundafixedaxis . . . . 156 7.4 InvarianceofthecompatibleVVFswithrespecttothechoiceofthe parameterization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 157 7.5 Perfectjoints . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 158 7.5.1 Definitionofaperfectjoint . . . . . . . . . . . . . . . . . . . . . . . . . . 158 7.5.2 Example1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 160 7.5.3 Example2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 161 7.5.4 Example3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 162 7.5.5 Example4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 165 7.6 Example:aperfectcompoundjoint . . . . . . . . . . . . . . . . . . . . . . . . . 167 7.6.1 Perfectcombinedjoint . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 168 7.6.2 Superimpositionoftwoperfectelementaryjoints . . . . . . . . . . . . . . . 169 8 Lagrange’sEquationsintheCaseofPerfectJoints 173 8.1 Lagrange’sequationsinthecaseofperfectjointsandanindependent parameterization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 174 8.1.1 Lagrange’sequations. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 174 8.1.2 Review . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 174 8.1.3 Particularcase . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 175 8.2 Lagrange’sequationsinthecaseofperfectjointsandinthepresenceof complementaryconstraintequations . . . . . . . . . . . . . . . . . . . . . . . . . 175 8.2.1 Lagrange’sequationswithmultipliers . . . . . . . . . . . . . . . . . . . . . 176 8.2.2 PracticalcalculationusingLagrange’smultipliers . . . . . . . . . . . . . . 178 8.2.3 Review . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 180 8.2.4 Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 180 8.3 Example:particleonarotatinghoop . . . . . . . . . . . . . . . . . . . . . . . . 181 8.3.1 Independentparameterization . . . . . . . . . . . . . . . . . . . . . . . . . 182 8.3.2 Reducedparameterizationno.1 . . . . . . . . . . . . . . . . . . . . . . . . 183 8.3.3 Reducedparameterizationno.2 . . . . . . . . . . . . . . . . . . . . . . . . 184 8.3.4 Calculationoftheenginetorque . . . . . . . . . . . . . . . . . . . . . . . . 185 8.4 Example:rigidbodyconnectedtoarotatingrodbyasphericaljoint(no.1) . . . . 187 8.5 Example:rigidbodyconnectedtoarotatingrodbyaspherical(jointno.2) . . . . 189 8.5.1 Totalparameterization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 189 8.5.2 Independentparameterization . . . . . . . . . . . . . . . . . . . . . . . . . 191 8.6 Example:rigidbodysubjectedtoadoublecontact . . . . . . . . . . . . . . . . . 191 8.6.1 Preliminaryanalysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 192 8.6.2 Independentparameterization . . . . . . . . . . . . . . . . . . . . . . . . . 194 8.6.3 Reducedparameterization . . . . . . . . . . . . . . . . . . . . . . . . . . . 196 Contents ix 9 FirstIntegrals 199 9.1 Painlevé’sfirstintegral . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 199 9.1.1 Painlevé’slemma. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 199 9.1.2 Painlevé’sfirstintegral . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 201 9.2 Theenergyintegral:conservativesystems. . . . . . . . . . . . . . . . . . . . . . 203 9.2.1 Energyconsiderationsinadditiontotheenergyintegral . . . . . . . . . . . 204 9.3 Example:diskrollingonasuspendedrod . . . . . . . . . . . . . . . . . . . . . . 205 9.4 Example:particleonarotatinghoop . . . . . . . . . . . . . . . . . . . . . . . . 208 9.5 Example:arigidbodyconnectedtoarotatingrodbyasphericaljoint(no.1) . . . 208 9.5.1 FirstintegralsviaNewtonianmechanics . . . . . . . . . . . . . . . . . . . 209 9.6 Example:rigidbodyconnectedtoarotatingrodbyasphericaljoint(no.2) . . . . 209 9.7 Example:rigidbodysubjectedtoadoublecontact . . . . . . . . . . . . . . . . . 210 9.7.1 UsingNewtonianmechanicstofindafirstintegral . . . . . . . . . . . . . . 211 10 Equilibrium 213 10.1Definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 213 10.1.1 Absoluteequilibrium. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 213 10.1.2 Parametricequilibrium. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 214 10.2Equilibriumequations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 215 10.2.1 Listofequationsandunknowns . . . . . . . . . . . . . . . . . . . . . . . . 218 10.2.2 Theexplicitpresenceoftimeinequilibriumequations . . . . . . . . . . . . 218 10.3Equilibriumequationsinthecaseofperfectjointsandindependent parameterization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 219 10.3.1 Listofequationsandunknowns . . . . . . . . . . . . . . . . . . . . . . . . 220 10.4Equilibriumequationsinthecaseofperfectjointsandinthepresenceof complementaryconstraintequations . . . . . . . . . . . . . . . . . . . . . . . . . 221 10.4.1 Listofequationsandunknowns . . . . . . . . . . . . . . . . . . . . . . . . 222 10.5Stabilityofanequilibrium . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 223 10.6Example:equilibriumofajack . . . . . . . . . . . . . . . . . . . . . . . . . . . 224 10.7Example:equilibriumofaliftingplatform . . . . . . . . . . . . . . . . . . . . . 225 10.8Example:equilibriumofarodinagutter . . . . . . . . . . . . . . . . . . . . . . 227 10.9Example:existenceofrangesofequilibriumpositions . . . . . . . . . . . . . . . 229 10.10Example:relativeequilibriumwithrespecttoarotatingreferenceframe . . . . . 230 10.11Example:equilibriuminthepresenceofcontactinequalities . . . . . . . . . . . 232 10.12Calculatinginternalefforts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 236 10.13Example:internaleffortsinatruss . . . . . . . . . . . . . . . . . . . . . . . . . 236 10.13.1 TensionforceinbarA(cid:3)A . . . . . . . . . . . . . . . . . . . . . . . . . . . 236 10.13.2 TensionforceinbarB(cid:3)B . . . . . . . . . . . . . . . . . . . . . . . . . . . 238 10.14Example:internaleffortsinatripod . . . . . . . . . . . . . . . . . . . . . . . . 240 11 RevisionProblems 243 11.1Equilibriumoftworods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 243 11.1.1 AnalysisusingNewton’slaw . . . . . . . . . . . . . . . . . . . . . . . . . 244 11.2Equilibriumofanelasticchair . . . . . . . . . . . . . . . . . . . . . . . . . . . . 245 11.3Equilibriumofadumptruck . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 246 11.4Equilibriumofasetsquare. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 248 11.5Motionofametronome . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 250 11.5.1 Equationofmotion. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 250 11.5.2 Firstintegral . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 251 11.5.3 Thecaseofsmalloscillations . . . . . . . . . . . . . . . . . . . . . . . . . 251 11.5.4 CasewherethebaseSmayslidewithoutfrictiononthetableT . . . . . . . 251

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Lagrangian Mechanics: An Advanced Analytical Approach PDF

318 Pages·2019·4.73 MB·English
by  Anh Le Van, Rabah Bouzidi
#Physics #Mechanics
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Author:Anh Le Van, Rabah Bouzidi
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