Classical Mechanics: Including an Introduction to the Theory of Elasticity PDF
Preview Classical Mechanics: Including an Introduction to the Theory of Elasticity
Undergraduate Lecture Notes in Physics Reinhard Hentschke Classical Mechanics Including an Introduction to the Theory of Elasticity Undergraduate Lecture Notes in Physics Undergraduate Lecture Notes in Physics (ULNP) publishes authoritative texts covering topicsthroughoutpureandappliedphysics.Eachtitleintheseriesissuitableasabasisfor undergraduateinstruction,typicallycontainingpracticeproblems,workedexamples,chapter summaries, andsuggestions for further reading. ULNP titles mustprovide at least oneof thefollowing: (cid:129) Anexceptionally clear andconcise treatment ofastandard undergraduate subject. (cid:129) Asolidundergraduate-levelintroductiontoagraduate,advanced,ornon-standardsubject. (cid:129) Anovel perspective oranunusual approach toteaching asubject. ULNPespeciallyencouragesnew,original,andidiosyncraticapproachestophysicsteaching at theundergraduate level. ThepurposeofULNPistoprovideintriguing,absorbingbooksthatwillcontinuetobethe reader’spreferred reference throughout theiracademic career. Series editors Neil Ashby University of Colorado, Boulder, CO, USA William Brantley Department of Physics, Furman University, Greenville, SC, USA Matthew Deady Physics Program, Bard College, Annandale-on-Hudson, NY, USA Michael Fowler Department of Physics, University of Virginia, Charlottesville, VA, USA Morten Hjorth-Jensen Department of Physics, University of Oslo, Oslo, Norway Michael Inglis SUNY Suffolk County Community College, Long Island, NY, USA Heinz Klose Humboldt University, Oldenburg, Niedersachsen, Germany Helmy Sherif Department of Physics, University of Alberta, Edmonton, AB, Canada More information about this series at http://www.springer.com/series/8917 Reinhard Hentschke Classical Mechanics Including an Introduction to the Theory of Elasticity 123 Reinhard Hentschke SchoolofMathematicsandNaturalSciences Bergische Universität Wuppertal Germany ISSN 2192-4791 ISSN 2192-4805 (electronic) Undergraduate Lecture Notesin Physics ISBN978-3-319-48709-0 ISBN978-3-319-48710-6 (eBook) DOI 10.1007/978-3-319-48710-6 LibraryofCongressControlNumber:2016958492 ©SpringerInternationalPublishingAG2017 Thisworkissubjecttocopyright.AllrightsarereservedbythePublisher,whetherthewholeorpart of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilms or in any other physical way, and transmission orinformationstorageandretrieval,electronicadaptation,computersoftware,orbysimilarordissimilar methodologynowknownorhereafterdeveloped. The use of general descriptive names, registered names, trademarks, service marks, etc. in this publicationdoesnotimply,evenintheabsenceofaspecificstatement,thatsuchnamesareexemptfrom therelevantprotectivelawsandregulationsandthereforefreeforgeneraluse. The publisher, the authors and the editors are safe to assume that the advice and information in this book are believed to be true and accurate at the date of publication. Neither the publisher nor the authorsortheeditorsgiveawarranty,expressorimplied,withrespecttothematerialcontainedhereinor foranyerrorsoromissionsthatmayhavebeenmade. Printedonacid-freepaper ThisSpringerimprintispublishedbySpringerNature TheregisteredcompanyisSpringerInternationalPublishingAG Theregisteredcompanyaddressis:Gewerbestrasse11,6330Cham,Switzerland Preface This textbook on classical mechanics is intended for physics students, who encounter the subject as a part of their undergraduate curriculum in theoretical physics. Chapter 1, Mathematical Background, reviews the mathematical ‘tool chest’ of classical mechanics. Emphasis is placed on the practical application of each tool making its usefulness to the subject as transparent as possible. Readers who are thoroughly familiar with the material may skip this chapter. But even then it is probably a good idea to at least briefly look over the different sections, focussing particularly on the problems and examples, where the mathematical ‘tools’ are applied in physics related contexts. Chapter 2, Laws of Mechanics, provides a general overview. It is intended as a guide rail for the beginner leading the way through the basic equations and concepts of mechanics. Chapter 3, Least Action Principle for one Coordinate, introduces Langrangian mechanics in simplified fashion.Accordingtotheauthor’sexperiencemostundergraduatephysicsstudents dofinditdifficulttointerrelatethestandardingredientsofundergraduatetheoretical physics, i.e. Newton’s equations of motion in mechanics, Maxwell’s equations in electrodynamics or Schrödinger’s equation in quantum mechanics. This, to me, makesitworthwhiletointroducetheleastactionprinciple,asaunifyingconceptin physics, at this early stage. Together the three chapters constitute the introductory part of the present text. The following five chapters are comprised of topics central to analytical mechanics.First,inChap.4,TheLeastActionPrinciple,theleastactionprincipleis reintroduced and discussed from a more general perspective. This includes the relation between conservation laws and symmetries, the description of motion in accelerated coordinate systems or dynamic stability. Chapter 5, Integration of the Equations of Motion, is devoted mainly to two-body problems including celestial mechanics and scattering. The next chapter, Small Oscillations, focuses on oscil- lations. This encompasses the standard harmonic oscillator including dissipation andexternalforces,dispersionrelations ofharmonicchains,normalmode analysis and related aspects. Chapter 7, Motion of Rigid Bodies, discusses the motion of rigid bodies. The moment of inertia tensor is defined and its meaning in different v vi Preface coordinate systems is discussed in detail. The equations of motion for rigid bodies are introduced including their representation in terms of the Euler equations using Euler angles and quaternions. The final chapter of the central part, Canonical Mechanics, introduces Hamiltonian dynamics and Hamilton-Jacobi theory. Thesubsequent chapters addresstopicsoutside thestandard contentofclassical mechanics.Chapter9,Many-BodySystems,focusesonthemechanicsofmany-body systems. This includes the numerical solution of the equations of motion, in par- ticulartheMolecularDynamicssimulationtechnique,aswellasthefoundationsof statistical mechanics contrasting the approaches due to Boltzmann and Gibbs. The chapter concludes with a brief discussion of the transition to chaos. Chapter 10, TheoryofElasticity,presentsanintroductiontothetheoryofelasticity.Thissubject, which is hardly ever touched upon in the current physics curriculum, possesses a widespectrumofapplications.Theserangefromthemicroscopicphysicsofcellsto thedesignofprecisioninstrumentationorthemacroscopicmechanicsofmaterials. Inmyopinion thetheoryofelasticity doesdeserveincreasedattention.The aimof thischapteristoprovidethestudentswithabasicintroductiontothissubjectandthe background knowledge on which to expand as necessary using more advanced literature.Atthebeginning,strain-andstresstensors,thefreeenergyofanisotropic elastic body and its attendant equilibrium conditions are derived and their uses are illustrated in a series of examples. Because in practice most problems in elasticity require numerical solution and because the method is of general use, Chap. 10 includes a section on the finite element method. The final section of the chapter summarizes the basic application of mechanics to viscoelasticity. This section also highlights that friction or, more generally, the dissipation of energy in mechanical systems is still not very well understood and continues to be a field of active research. Throughout the book the reader will encounter three types of highlighted examples.Mostofthemaresolvedproblems.Ifinditimportanttosupplythereader withopportunitiesforexercisingnewlyacquiredconceptswithoutgettingstuckand thusloosinginterest.Occasionallyaproblemreoccurs,whenitisusefultocompare different solution approaches. Then there are ordinary examples designed to prac- tice a new concept during reading. The only exception from this rule are the more elaborate numerical Mathematica examples in Chap. 10. In addition, a certain numberofadvancedexamplesservethepurposetoeitherexplainadifficultpointor relate the current material to other areas in physics. This can be an extension of classical mechanics to quantum mechanics, via the so-called quasi-classical approximation,orthestrengtheningofrubbermaterialsthroughfillernanoparticles. Theseexamples,eventhoughtheycanbeomittedonafirstreading,areintendedto supply additional motivation for the context in which they are embedded. Selectedheadingsareaccompaniedbyaraisedsymbol,whichprovidesarough guidance for the materials selection. The meaning of the symbols is as follows: (cid:129) †: Material of particular importance to the beginner. (cid:129) ‡: The content should be included according to time and necessity. (cid:129) no symbol: Materials important for more advanced students. Preface vii (cid:129) ~: The chapter on the theory of elasticity certainly requires considerable extra time. This extra time may not always be affordable. Thus, it is a ‘matter of the heart’ how much effort one is willing to spend on the various sections com- prising this chapter. Itisverylikelythatdespitemyefforttothecontrary,thistextwillcontainerrors. On my website (http://constanze.materials.uni-wuppertal.de) readers can find a continuously updated list of corrections. Wuppertal, Germany Reinhard Hentschke Contents 1 Mathematical Tools .... ..... .... .... .... .... .... ..... .... 1 1.1 Coordinatesy . .... ..... .... .... .... .... .... ..... .... 1 1.2 Vectorsy . .... .... ..... .... .... .... .... .... ..... .... 3 1.3 Matricesy .... .... ..... .... .... .... .... .... ..... .... 12 1.4 Derivatives and Integralsy .... .... .... .... .... ..... .... 18 1.5 Complex Numbersy ..... .... .... .... .... .... ..... .... 36 Reference . .... .... .... ..... .... .... .... .... .... ..... .... 38 2 Laws of Mechanics . .... ..... .... .... .... .... .... ..... .... 39 2.1 An Overviewy .... ..... .... .... .... .... .... ..... .... 39 2.2 Two Examples in Newtonian Mechanicsy .... .... ..... .... 53 References .... .... .... ..... .... .... .... .... .... ..... .... 68 3 Least Action Principle for One Coordinate .. .... .... ..... .... 69 3.1 Euler–Lagrange Equation for One Coordinatey .... ..... .... 69 3.2 Two Simple Examplesy .. .... .... .... .... .... ..... .... 73 3.3 The Meaning of the Least Action Principley .. .... ..... .... 75 Reference . .... .... .... ..... .... .... .... .... .... ..... .... 87 4 Principle of Least Action ..... .... .... .... .... .... ..... .... 89 4.1 Lagrangian for a System of Point Masses .... .... ..... .... 89 4.2 Conserved Quantities .... .... .... .... .... .... ..... .... 102 4.3 Lagrangians in Accelerated Systems .... .... .... ..... .... 106 4.4 An Application in Theoretical Chemistry .... .... ..... .... 119 References .... .... .... ..... .... .... .... .... .... ..... .... 122 5 Integrating the Equations of Motion .... .... .... .... ..... .... 123 5.1 One-Dimensional Motiony .... .... .... .... .... ..... .... 123 5.2 Two-Body Central Force Motiony .. .... .... .... ..... .... 125 ix x Contents 5.3 Scatteringz ... .... ..... .... .... .... .... .... ..... .... 141 References .... .... .... ..... .... .... .... .... .... ..... .... 153 6 Small Oscillations .. .... ..... .... .... .... .... .... ..... .... 155 6.1 One-Dimensional Motiony .... .... .... .... .... ..... .... 155 6.2 Normal Mode Analysis .. .... .... .... .... .... ..... .... 177 References .... .... .... ..... .... .... .... .... .... ..... .... 187 7 Rigid Body Motion . .... ..... .... .... .... .... .... ..... .... 189 7.1 Moment of Inertia Tensor and Angular Momentumy ..... .... 189 7.2 Equations of Motion for a Rigid Body .. .... .... ..... .... 209 7.3 Static Contact Between Rigid Bodiesy ... .... .... ..... .... 225 8 Canonical Mechanics ... ..... .... .... .... .... .... ..... .... 233 8.1 Hamilton’s Equations of Motion ... .... .... .... ..... .... 233 8.2 Hamilton–Jacobi Theory . .... .... .... .... .... ..... .... 246 9 Many-Particle Mechanics .... .... .... .... .... .... ..... .... 253 9.1 Numerical Solution of the Equations of Motiony ... ..... .... 253 9.2 Molecular Dynamics Simulation ... .... .... .... ..... .... 258 9.3 From Mechanics to Statistical Mechanicsz ... .... ..... .... 271 9.4 Classification of Dynamical Systems .... .... .... ..... .... 284 9.5 Roads to Chaos ... ..... .... .... .... .... .... ..... .... 287 References .... .... .... ..... .... .... .... .... .... ..... .... 289 10 Basic Equations of the Theory of Elasticity~ . .... .... ..... .... 291 10.1 Strain and Stress Tensors .... .... .... .... .... ..... .... 292 10.2 Free Energy .. .... ..... .... .... .... .... .... ..... .... 301 10.3 Examples .... .... ..... .... .... .... .... .... ..... .... 314 10.4 Finite Element Method .. .... .... .... .... .... ..... .... 332 10.5 Dynamic Mechanical Analysis of Viscoelastic Materials .. .... 352 References .... .... .... ..... .... .... .... .... .... ..... .... 364 Appendix A: Identities and Units... .... .... .... .... .... ..... .... 365 Appendix B: Mathematica MD in the NVE-Ensemble .. .... ..... .... 369 Index .... .... .... .... .... ..... .... .... .... .... .... ..... .... 375
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