Undergraduate Lecture Notes in Physics Carl S. Helrich Analytical Mechanics 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 ofPhysics, FurmanUniversity, Greenville, SC,USA MatthewDeady Physics Program, BardCollege, Annandale-on-Hudson, NY,USA Michael Fowler Department ofPhysics, University of Virginia, Charlottesville, VA, USA Morten Hjorth-Jensen Department ofPhysics, University of Oslo, Oslo, Norway Michael Inglis SUNY SuffolkCountyCommunity College, LongIsland, NY,USA Heinz Klose Humboldt University, Oldenburg,Niedersachsen, Germany HelmySherif Department ofPhysics, University of Alberta, Edmonton, AB,Canada More information about this series at http://www.springer.com/series/8917 Carl S. Helrich Analytical Mechanics 123 Carl S. Helrich Goshen,IN USA Additional material tothis bookcanbedownloaded from http://extras.springer.com. ISSN 2192-4791 ISSN 2192-4805 (electronic) Undergraduate Lecture Notesin Physics ISBN978-3-319-44490-1 ISBN978-3-319-44491-8 (eBook) DOI 10.1007/978-3-319-44491-8 LibraryofCongressControlNumber:2016948228 ©SpringerInternationalPublishingSwitzerland2017 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 To my wife, for her patience and understanding Preface This textbook presents what Joseph-Louis Lagrange called Analytical Mechanics. Historically this was a great advance beyond the methods of Euclidean geometry employed by Isaac Newton in the Philosophiae Naturalis Principia Mathematica. With the methods of Lagrange and Leonhard Euler, we could actually perform calculations. Lagrange and Euler used the calculus and did not require the for- midable expertise in the use of geometry that Newton possessed. The step introduced by William Rowan Hamilton simplified the formulation. Hamilton’s ideas also represent a great step forward in our understanding of the meaning of Analytical Mechanics. This, coupled with the simplification added by Carl Gustav Jacobi, provided us with a pathway to the more modern uses of AnalyticalMechanicsincludingapplicationstoastrophysics,complexsystems,and chaos. Ourapproachwillintroduceamodernversionofwhatwasdoneinthe18thand 19th centuries. We will follow essentially the historical development because the ideasunfoldmostlogicallyifwedoso.Wewill,however,paymoreattentiontothe development of Analytical Mechanics as a valuable tool than to a historical study. Our final step will be the relativistic formulation of Analytical Mechanics. That is an absolute necessity in any complete study of Analytical Mechanics. Logically we begin this text with a chapter on the history of mechanics. Many texts include brief historical comments or even added pages outlining individual contributions.Thatiscertainlyanimprovementontheanecdotesthatourprofessors often passed on to us without citation. Those anecdotes piqued our interest and added flavor. But they lacked a continuity of thought and that all-important accu- racy that we prize. Analytical Mechanics is the oldest of the sciences. And the history stretches from the beginnings of philosophy in Miletus in 600 BCE to the advances in scientific thought introduced in the Prussian Academy and in Great Britain. I have sincerely endeavored to shorten this, as any serious student will easily recognize. But I still worry about the length. vii viii Preface Because myown understanding ofsciencehasbeen greatly enriched by studies inhistory,Icannotrecommendthataprofessorignorethefirstchaptercompletely. Thestudentshouldunderstandsomethingoftheinterestingandtorturedindividual Newton was. And we cannot really comprehend the origins of the ideas that gave birthtoAnalyticalMechanicswithoutencounteringtheworkofPierreMaupertuis, Johann Bernoulli,1Euler, andLagrange. ThesectionsofHamilton andJacobi may be held until after the students have gained an appreciation for the methods of Analytical Mechanics. But those sections will be of interest to students as they encounter the chapter on the Hamilton-Jacobi approach. They should see the simplicity of what Jacobi brought and his great respect for the ideas of Hamilton. Then to emphasize the importance these ideas, I include an outline of Erwin Schrödinger’s original published derivation of his wave equation from the Hamilton-Jacobi equation. With the caveat surrounding a second variation, the quantum theory is buried in the theory of Hamilton and Jacobi. InChap.1,IhavenotincludedthehistoricaleventsleadingtoAlbertEinstein’s developmentoftheSpecialTheoryofRelativityin1905.SomeofthisIhaveplaced in the final chapter. The historical importance of Einstein’s contributions is more easily understood by a reader who has a general grasp of the classical theory of fields, which is not our primary topic. Beyondthehistory,theprimarypartofthetext,inwhichIpresentthebasisand applications of Analytical Mechanics, begins with Chap. 2 on Lagrangian Mechanics. There the issue is the Euler-Lagrange equation and the variational problem, which is solved by the Euler-Lagrange equation. This I follow by a chapteronHamiltonianMechanics,which,throughtheLegendretransformation,is a logical next step from the approach of Euler and Lagrange. The canonical equationswereactuallyobtainedbyHamiltoninhispapersof1834and1835with another goal in mind. But the procedure was the Legendre transformation. With these chapters, we have Analytical Mechanics essentially in place. ThenIintroducetheHamilton-Jacobiapproach.Idonotpresentamethodtobe memorized and applied because doing so obscures the logic and the simplicity. I follow in spirit, but not in precise detail, the ideas of Jacobi. The generator of a canonical transformation will take center stage, as it didfor him.The final method does not follow a head-down approach, but one with finesse. In all texts there are final chapters. And all courses are of finite duration. Therefore, there will always be parts of the student’s experience that will become lost in the fuzziness of the final days. In this text, those final chapters contain studies of complex systems, chaos, and relativistic mechanics. Each of these chaptersdealswithsubjectsofentirecoursesatmanyinstitutions.Ihavewrittenthe chaptersoncomplexsystemsandonchaosasintroductionstotheseveryinteresting topics.Theymaythenbetreatedaswindowsopeningontostudiesthatmayoccupy 1Johannwastheoriginalnamegivenbyhisparents.JeanorJohnappearssometimes,depending upon whether the author is French or English. Johann Bernoulli was born and died in Basel, Switzerland. Preface ix thestudents’interestscompletelyatalatertime.Theymayevenprovideinterestfor thelastweeksofasemester.Butthefinalchapteronspecialrelativityisnotofthe same character. I elected deliberately to make the final chapter on the Special Theory of Relativityanalmostself-containedunit.Thereaderwhoisnotcompletelyfamiliar withthetheoryofclassicalfieldswillbeabletopassoveraportionofthechapterin whichwedevelopthefieldstrengthtensorandelectromagneticforce.However,the approach to relativistic mechanics and finally to relativistic Analytical Mechanics should be considered carefully by the serious reader. There I have followed some of the classic sources, such as Wolfgang Pauli, Peter G. Bergmann, and Wolgang Rindler. The principal product of this work is the Hamiltonian and the canonical equations for relativistic motion in the electromagnetic field. We required the nonrelativistic approximation to these results for our treatment of this motion in a previous chapter. Icannotexpectthatallstudentswillbestirred,assomeofminehavebeen,when theyseetheconnectionsamongtheideascommontotheoreticalphysics.ButIhope they are. I am deeply indebted to generations of students who have gone through this intellectualadventurewithmeduringthepastfortyyears.IamthankfulthatIhave been part of their intellectual pathways and for the questions with which they continued to press me. They have seen me grow in understanding and love for the ideas I try to express here. I am also very thankful to my teachers who introduced me to the beauty and power of Analytical Mechanics. Isaac Greber particularly stands out. He presented us with remarkably inspired and almost impossibly difficult problems to which I devotedallofmyenergiesonmanycoldwinternightsinCleveland.Isaachasbeen a friend and an inspiration. Goshen, IN, USA Carl S. Helrich May 2016 Contents 1 History.... .... .... .... ..... .... .... .... .... .... ..... .... 1 1.1 Introduction ... .... ..... .... .... .... .... .... ..... .... 1 1.2 Ancient Greece. .... ..... .... .... .... .... .... ..... .... 2 1.2.1 Milesians... ..... .... .... .... .... .... ..... .... 3 1.2.2 Beyond Miletus... .... .... .... .... .... ..... .... 4 1.2.3 Atomism ... ..... .... .... .... .... .... ..... .... 7 1.2.4 Plato .. .... ..... .... .... .... .... .... ..... .... 8 1.2.5 Aristotle.... ..... .... .... .... .... .... ..... .... 9 1.2.6 Reflection on Greek Science. .... .... .... ..... .... 11 1.3 Islamic Science. .... ..... .... .... .... .... .... ..... .... 11 1.3.1 The Rise of Islamic Science . .... .... .... ..... .... 12 1.3.2 Expansion and Contact . .... .... .... .... ..... .... 12 1.3.3 Islam and Greek Astronomy . .... .... .... ..... .... 13 1.3.4 Islamic Astronomy .... .... .... .... .... ..... .... 14 1.3.5 Decline of Islamic Science .. .... .... .... ..... .... 16 1.3.6 Reflection on Islamic Science .... .... .... ..... .... 17 1.4 Europe Encounters Islamic Science .. .... .... .... ..... .... 17 1.5 Medieval Physics ... ..... .... .... .... .... .... ..... .... 18 1.6 European Scientific Revolution.. .... .... .... .... ..... .... 20 1.7 Newton... .... .... ..... .... .... .... .... .... ..... .... 21 1.7.1 Introduction. ..... .... .... .... .... .... ..... .... 21 1.7.2 The Person . ..... .... .... .... .... .... ..... .... 21 1.7.3 Disputes ... ..... .... .... .... .... .... ..... .... 24 1.7.4 Principia ... ..... .... .... .... .... .... ..... .... 26 1.7.5 Newtonian Concepts ... .... .... .... .... ..... .... 27 1.8 Eighteenth Century.. ..... .... .... .... .... .... ..... .... 31 1.8.1 Maupertuis . ..... .... .... .... .... .... ..... .... 31 1.8.2 Euler.. .... ..... .... .... .... .... .... ..... .... 33 1.8.3 Lagrange... ..... .... .... .... .... .... ..... .... 35 xi