Mechanics: An Intensive Course

476 Pages·2012·2.99 MB·English

by  Masud Chaichian, Ioan Merches, Anca Tureanu (auth.)

Mechanics
Masud Chaichian Ioan Merches
•
Anca Tureanu
Mechanics
An Intensive Course
123
Prof.Dr. MasudChaichian Prof.Dr. IoanMerches
Department of Physics Faculty ofPhysics
Universityof Helsinki Al.I.Cuza University
PO Box64 CarolI Boulevard11
00014Helsinki 700506Iasi
Finland Romania
e-mail: masud.chaichian@helsinki.fi e-mail: [email protected]
Doc. Dr.Anca Tureanu
Department of Physics
Universityof Helsinki
PO Box64
00014Helsinki
Finland
e-mail: anca.tureanu@helsinki.fi
ISBN 978-3-642-16390-6 e-ISBN978-3-642-17234-2
DOI 10.1007/978-3-642-17234-2
SpringerHeidelbergDordrechtLondonNewYork
(cid:2)Springer-VerlagBerlinHeidelberg2012
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He who knows nothing, loves nothing. But he
who understands also loves, notices, sees.
The more knowledge is involved in a thing,
the greater the love.
Paracelsus
To our parents and to their memory
Preface
Mechanics is the oldest discipline among the fundamental natural sciences. The
namecomesfromtheGreekword‘‘mechanike’’,whichmeans‘‘mechanism’’.The
subject of mechanics asa science is the investigation ofthe motionofbodiesand
theirequilibriumundertheactionofappliedforces.Dependingonthenatureofthe
bodies, mechanics can be divided into three branches: (a) general mechanics,
dealingwiththemechanicalbehaviourofmaterialpointsandrigidbodies;(b)fluid
mechanics(orthemechanicsofcontinuousmedia),whichisconcernedwithideal
and viscous fluids and (c) mechanics of deformable media, which studies the
deformation of solid bodies under applied external forces.
The knowledge of mechanical motion or displacement of bodies can be
accomplished by a very general procedure based on a system of basic axioms,
called principles. These principles are the core of what is known as Newtonian
mechanics, relativistic mechanics, quantum mechanics and so forth. During the
eighteenth century, after the huge success achieved by the mechanics of Galileo
Galilei (1564–1642) and Isaac Newton (1643–1727), there appeared the tendency
of making mechanics more abstract and general. This tendency leads to what
nowadays is called analytical mechanics. Among the founders of analytical
mechanicsare: Pierre-Louis MoreaudeMaupertuis (1698–1759), Leonhard Euler
(1707–1783), Jean Baptiste le Rond D’Alembert (1717–1783), Joseph-Louis
Lagrange (1736–1813), Carl Friedrich Gauss (1777–1855) and William Rowan
Hamilton (1805–1865). Analytical mechanics has proved to be a very useful tool
of investigation not only in Newtonian mechanics, but also in other disciplines of
Physics:electrodynamics, quantum field theory,theoryofrelativity,magnetofluid
dynamics – to mention a few.
Classical mechanics has undergone an important revival during the last few
decades, due to the progress in non-linear dynamics, stochastic processes and
various applications of Noether’s theorem in the study of both discrete and con-
tinuoussystems.WerecallthattherearenoexactlylinearprocessesinNature,but
only approximately. All linear models studied in any science are only approxi-
mations of reality.
ix
x Preface
Thisbookisdedicatedtotheprinciplesandapplicationsofclassicalmechanics,
writtenforundergraduateandgraduatestudentsinphysicsandrelatedsubjects.Its
mainpurposeistomakethestudentsfamiliarwiththefundamentalsofthetheory,
to stimulate them in the use of applications and to contribute to the formation of
their background as specialists.
The first two chapters are dedicated to the basic notions and principles of both
Newtonianandanalyticalmechanics,asdifferentapproachestothesamepurpose:
theinvestigationofmechanicalbehaviourofbothdiscreteandcontinuoussystems.
A special emphasis is put on the large applicability of analytical formalism in
various branches of physics.
In the third chapter, the Lagrangian formalism is applied to the study of some
classic mechanical systems, as the harmonic oscillator and the gravitational pen-
dulum, as well as to the investigation of some non-mechanical systems, like
electric circuits.
The fourth chapter is concerned with the mechanics of the rigid body. The
derivation of velocity and acceleration distributions in relative motion makes
possibletostudythemotionofarigid bodyaboutafixedpoint.Thechapterends
with some applications, such as the physical pendulum and the symmetrical top,
together with some mechanical–electromagnetic analogies.
TheaimofthefifthchapteristomakethereaderfamiliarwiththeHamiltonian
formalism. The derivation of the canonical equations is followed by several
applications and extensions in mechanics and electrodynamics. The canonical
transformations, integral invariants and the Hamilton–Jacobi formalism are also
described. They are very useful for students for their further studies of thermo-
dynamics, statistics and quantum theory.
The sixth, final, chapter deals with the mechanics of continuous deformable
media.Here,boththeLagrangianandHamiltonianformalismsareappliedinorder
to study some well-known models of continuous media: the elastic medium, the
ideal and viscous fluids. Special attention is paid to the extension of Noether’s
theorem to continuous media and its applications to the fundamental theorems of
ideal fluids.
Since classical mechanics has undergone a considerable evolution during the
lastcentury,theauthorshavetriedtodrawtheattentionofthereadertothreemain
directions of development of post-classical mechanics: theory of relativity,
quantum mechanics and stochastic processes. These three basic orientations in
post-classical mechanics are very briefly exposed in three addenda, which con-
cludethemainsubstanceofthebook.Attheendofthebook,fortheconvenience
of readers, two appendices are provided, which contain the most frequently used
formulas on vector and tensor algebra, as well as on vector calculus.
Thepresentbookisanoutcomeoftheauthors’teachingexperienceovermany
years in different countries and with different students studying diverse fields of
physics and engineering. The authors believe that the presentation and the dis-
tributionofthe topics, thevariousapplications inseveralbranches ofphysics and
thesetofmorethan100proposedproblemsmakethisbookacomprehensiveand
useful tool for students, teachers and researchers.
Preface xi
During thepreparationofthisbooktheauthorshave benefitedfromdiscussing
various questions with many of their colleagues and students. It is a pleasure to
express gratitude to all of them and to acknowledge the stimulating discussions
and their useful advice. Our special thanks go to Professor Peter Presnajder for
valuable suggestions and for his considerable help in improving the manuscript.
Helsinki, Iasi, October 2011 M. Chaichian
I. Merches
A. Tureanu
Contents
1 Foundations of Newtonian Mechanics. . . . . . . . . . . . . . . . . . . . . . 1
1.1 Notions, Principles and Fundamental Theorems
of Newtonian Mechanics. . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.1.1 Velocity. Acceleration . . . . . . . . . . . . . . . . . . . . . . . 2
1.1.2 Analytical Expressions for Velocity and Acceleration
in Different Coordinate Systems . . . . . . . . . . . . . . . . 4
1.2 Principles of Newtonian Mechanics . . . . . . . . . . . . . . . . . . . . 6
1.2.1 The Principle of Inertia (Newton’s First Law). . . . . . . 6
1.2.2 The Law of Force (Newton’s Second Law). . . . . . . . . 7
1.2.3 The Principle of Action and Reaction
(Newton’s Third Law) . . . . . . . . . . . . . . . . . . . . . . . 8
1.3 General Theorems of Newtonian Mechanics . . . . . . . . . . . . . . 9
1.3.1 Integration of the Equations of Motion. . . . . . . . . . . . 9
1.3.2 First Integrals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
1.3.3 General Theorems of One-Particle Mechanics. . . . . . . 12
1.3.4 General Theorems for Systems of Particles. . . . . . . . . 15
1.4 Problems. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23
2 Principles of Analytical Mechanics. . . . . . . . . . . . . . . . . . . . . . . . 27
2.1 Constraints . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27
2.1.1 One-Particle Systems . . . . . . . . . . . . . . . . . . . . . . . . 27
2.1.2 Many-Particle Systems. . . . . . . . . . . . . . . . . . . . . . . 39
2.2 Elementary Displacements . . . . . . . . . . . . . . . . . . . . . . . . . . 40
2.3 Principle of Virtual Work. . . . . . . . . . . . . . . . . . . . . . . . . . . 42
2.3.1 Free Particle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42
2.3.2 Particle Subject to Constraints. . . . . . . . . . . . . . . . . . 42
2.3.3 System of Free Particles. . . . . . . . . . . . . . . . . . . . . . 43
2.3.4 System of Particles Subject to Constraints . . . . . . . . . 43
2.3.5 Application. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46
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