Mechanics

Masud Chaichian • Ioan MerchesAnca Tureanu

Mechanics

An Intensive Course

123

Prof. Dr. Masud ChaichianDepartment of PhysicsUniversity of HelsinkiPO Box 6400014 HelsinkiFinlande-mail: masud.chaichian@helsinki.fi

Prof. Dr. Ioan MerchesFaculty of PhysicsAl. I. Cuza UniversityCarol I Boulevard 11700506 IasiRomaniae-mail: merches@uaic.ro

Doc. Dr. Anca TureanuDepartment of PhysicsUniversity of HelsinkiPO Box 6400014 HelsinkiFinlande-mail: anca.tureanu@helsinki.fi

ISBN 978-3-642-16390-6 e-ISBN 978-3-642-17234-2DOI 10.1007/978-3-642-17234-2Springer Heidelberg Dordrecht London New York

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Paracelsus

To our parents and to their memory

Preface

Mechanics is the oldest discipline among the fundamental natural sciences. Thename comes from the Greek word ‘‘mechanike’’, which means ‘‘mechanism’’. Thesubject of mechanics as a science is the investigation of the motion of bodies andtheir equilibrium under the action of applied forces. Depending on the nature of thebodies, mechanics can be divided into three branches: (a) general mechanics,dealing with the mechanical behaviour of material points and rigid bodies; (b) fluidmechanics (or the mechanics of continuous media), which is concerned with idealand viscous fluids and (c) mechanics of deformable media, which studies thedeformation of solid bodies under applied external forces.

The knowledge of mechanical motion or displacement of bodies can beaccomplished by a very general procedure based on a system of basic axioms,called principles. These principles are the core of what is known as Newtonianmechanics, relativistic mechanics, quantum mechanics and so forth. During theeighteenth century, after the huge success achieved by the mechanics of GalileoGalilei (1564–1642) and Isaac Newton (1643–1727), there appeared the tendencyof making mechanics more abstract and general. This tendency leads to whatnowadays is called analytical mechanics. Among the founders of analyticalmechanics are: Pierre-Louis Moreau de Maupertuis (1698–1759), Leonhard Euler(1707–1783), Jean Baptiste le Rond D’Alembert (1717–1783), Joseph-LouisLagrange (1736–1813), Carl Friedrich Gauss (1777–1855) and William RowanHamilton (1805–1865). Analytical mechanics has proved to be a very useful toolof investigation not only in Newtonian mechanics, but also in other disciplines ofPhysics: electrodynamics, quantum field theory, theory of relativity, magnetofluiddynamics – to mention a few.

Classical mechanics has undergone an important revival during the last fewdecades, due to the progress in non-linear dynamics, stochastic processes andvarious applications of Noether’s theorem in the study of both discrete and con-tinuous systems. We recall that there are no exactly linear processes in Nature, butonly approximately. All linear models studied in any science are only approxi-mations of reality.

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This book is dedicated to the principles and applications of classical mechanics,written for undergraduate and graduate students in physics and related subjects. Itsmain purpose is to make the students familiar with the fundamentals of the theory,to stimulate them in the use of applications and to contribute to the formation oftheir background as specialists.

The first two chapters are dedicated to the basic notions and principles of bothNewtonian and analytical mechanics, as different approaches to the same purpose:the investigation of mechanical behaviour of both discrete and continuous systems.A special emphasis is put on the large applicability of analytical formalism invarious branches of physics.

In the third chapter, the Lagrangian formalism is applied to the study of someclassic mechanical systems, as the harmonic oscillator and the gravitational pen-dulum, as well as to the investigation of some non-mechanical systems, likeelectric circuits.

The fourth chapter is concerned with the mechanics of the rigid body. Thederivation of velocity and acceleration distributions in relative motion makespossible to study the motion of a rigid body about a fixed point. The chapter endswith some applications, such as the physical pendulum and the symmetrical top,together with some mechanical–electromagnetic analogies.

The aim of the fifth chapter is to make the reader familiar with the Hamiltonianformalism. The derivation of the canonical equations is followed by severalapplications and extensions in mechanics and electrodynamics. The canonicaltransformations, integral invariants and the Hamilton–Jacobi formalism are alsodescribed. 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 deformablemedia. Here, both the Lagrangian and Hamiltonian formalisms are applied in orderto study some well-known models of continuous media: the elastic medium, theideal and viscous fluids. Special attention is paid to the extension of Noether’stheorem to continuous media and its applications to the fundamental theorems ofideal fluids.

Since classical mechanics has undergone a considerable evolution during thelast century, the authors have tried to draw the attention of the reader to three maindirections of development of post-classical mechanics: theory of relativity,quantum mechanics and stochastic processes. These three basic orientations inpost-classical mechanics are very briefly exposed in three addenda, which con-clude the main substance of the book. At the end of the book, for the convenienceof readers, two appendices are provided, which contain the most frequently usedformulas on vector and tensor algebra, as well as on vector calculus.

The present book is an outcome of the authors’ teaching experience over manyyears in different countries and with different students studying diverse fields ofphysics and engineering. The authors believe that the presentation and the dis-tribution of the topics, the various applications in several branches of physics andthe set of more than 100 proposed problems make this book a comprehensive anduseful tool for students, teachers and researchers.

x Preface

During the preparation of this book the authors have benefited from discussingvarious questions with many of their colleagues and students. It is a pleasure toexpress gratitude to all of them and to acknowledge the stimulating discussionsand their useful advice. Our special thanks go to Professor Peter Presnajder forvaluable suggestions and for his considerable help in improving the manuscript.

Helsinki, Iasi, October 2011 M. ChaichianI. Merches

A. Tureanu

Preface xi

Contents

1 Foundations of Newtonian Mechanics . . . . . . . . . . . . . . . . . . . . . . 11.1 Notions, Principles and Fundamental Theorems

of Newtonian Mechanics. . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.1.1 Velocity. Acceleration . . . . . . . . . . . . . . . . . . . . . . . 21.1.2 Analytical Expressions for Velocity and Acceleration

in Different Coordinate Systems . . . . . . . . . . . . . . . . 41.2 Principles of Newtonian Mechanics . . . . . . . . . . . . . . . . . . . . 6

1.2.1 The Principle of Inertia (Newton’s First Law) . . . . . . . 61.2.2 The Law of Force (Newton’s Second Law). . . . . . . . . 71.2.3 The Principle of Action and Reaction

(Newton’s Third Law) . . . . . . . . . . . . . . . . . . . . . . . 81.3 General Theorems of Newtonian Mechanics . . . . . . . . . . . . . . 9

1.3.1 Integration of the Equations of Motion . . . . . . . . . . . . 91.3.2 First Integrals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111.3.3 General Theorems of One-Particle Mechanics . . . . . . . 121.3.4 General Theorems for Systems of Particles . . . . . . . . . 15

1.4 Problems. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23

2 Principles of Analytical Mechanics . . . . . . . . . . . . . . . . . . . . . . . . 272.1 Constraints . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27

2.1.1 One-Particle Systems . . . . . . . . . . . . . . . . . . . . . . . . 272.1.2 Many-Particle Systems . . . . . . . . . . . . . . . . . . . . . . . 39

2.2 Elementary Displacements . . . . . . . . . . . . . . . . . . . . . . . . . . 402.3 Principle of Virtual Work . . . . . . . . . . . . . . . . . . . . . . . . . . . 42

2.3.1 Free Particle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 422.3.2 Particle Subject to Constraints . . . . . . . . . . . . . . . . . . 422.3.3 System of Free Particles . . . . . . . . . . . . . . . . . . . . . . 432.3.4 System of Particles Subject to Constraints . . . . . . . . . 432.3.5 Application . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46

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2.4 Generalized Coordinates . . . . . . . . . . . . . . . . . . . . . . . . . . . . 472.4.1 Configuration Space . . . . . . . . . . . . . . . . . . . . . . . . . 482.4.2 Generalized Forces. . . . . . . . . . . . . . . . . . . . . . . . . . 492.4.3 Kinetic Energy in Generalized Coordinates . . . . . . . . . 51

2.5 Differential and Integral Principles in Analytical Mechanics . . . 522.5.1 D’Alembert’s Principle . . . . . . . . . . . . . . . . . . . . . . . 532.5.2 Lagrange Equations for Holonomic Systems . . . . . . . . 532.5.3 Velocity-Dependent Potential . . . . . . . . . . . . . . . . . . 572.5.4 Non-potential Forces . . . . . . . . . . . . . . . . . . . . . . . . 60

2.6 Elements of Calculus of Variations . . . . . . . . . . . . . . . . . . . . 622.6.1 Shortest Distance Between Two Points in a Plane . . . . 662.6.2 Brachistochrone Problem . . . . . . . . . . . . . . . . . . . . . 672.6.3 Surface of Revolution of Minimum Area . . . . . . . . . . 682.6.4 Geodesics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69

2.7 Hamilton’s Principle. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 742.7.1 Euler–Lagrange Equations for the Action Integral . . . . 772.7.2 Criteria for the Construction of Lagrangians . . . . . . . . 79

2.8 Symmetry Properties and Conservation Theorems . . . . . . . . . . 792.8.1 First Integrals as Constants of Motion . . . . . . . . . . . . 802.8.2 Symmetry Transformations . . . . . . . . . . . . . . . . . . . . 822.8.3 Noether’s Theorem. . . . . . . . . . . . . . . . . . . . . . . . . . 82

2.9 Principle of Least Action . . . . . . . . . . . . . . . . . . . . . . . . . . . 862.10 Problems. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93

3 Applications of the Lagrangian Formalism in the Studyof Discrete Particle Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 973.1 Central Force Fields . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97

3.1.1 Two-Body Problem . . . . . . . . . . . . . . . . . . . . . . . . . 973.1.2 General Properties of Motion in Central Field . . . . . . . 993.1.3 Discussion of Trajectories . . . . . . . . . . . . . . . . . . . . . 1013.1.4 Bertrand’s Theorem . . . . . . . . . . . . . . . . . . . . . . . . . 105

3.2 Kepler’s Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1083.2.1 Determination of Trajectories . . . . . . . . . . . . . . . . . . 1093.2.2 Law of Motion . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1133.2.3 Runge–Lenz Vector . . . . . . . . . . . . . . . . . . . . . . . . . 1163.2.4 Artificial Satellites of the Earth. Cosmic Velocities . . . 118

3.3 Classical Theory of Collisions Between Particles . . . . . . . . . . . 1213.3.1 Collisions Between Two Particles . . . . . . . . . . . . . . . 1213.3.2 Effective Scattering Cross Section . . . . . . . . . . . . . . . 1253.3.3 Scattering on a Spherical Potential Well . . . . . . . . . . . 1293.3.4 Rutherford’s Formula . . . . . . . . . . . . . . . . . . . . . . . . 131

3.4 Periodical Motion of a Particle Under the Influence of Gravity . . . 1323.4.1 Simple Pendulum. . . . . . . . . . . . . . . . . . . . . . . . . . . 1333.4.2 Cycloidal Pendulum . . . . . . . . . . . . . . . . . . . . . . . . . 1363.4.3 Spherical Pendulum . . . . . . . . . . . . . . . . . . . . . . . . . 137

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3.5 Motion of a Particle Subject to an Elastic Force . . . . . . . . . . . 1403.5.1 Harmonic Linear Oscillator . . . . . . . . . . . . . . . . . . . . 1403.5.2 Space Oscillator. . . . . . . . . . . . . . . . . . . . . . . . . . . . 1413.5.3 Non-linear Oscillations . . . . . . . . . . . . . . . . . . . . . . . 143

3.6 Small Oscillations About a Position of Stable Equilibrium . . . . 1443.6.1 Equations of Motion. Normal Coordinates . . . . . . . . . 1463.6.2 Small Oscillations of Molecules. . . . . . . . . . . . . . . . . 149

3.7 Analogy Between Mechanical and Electric Systems . . . . . . . . . 1533.7.1 Kirchhoff’s Rule Relative to the Loops of an Electric

Circuit . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1533.7.2 Kirchhoff’s Rule Relative to the Junction Points

of an Electric Circuit . . . . . . . . . . . . . . . . . . . . . . . . 1573.8 Problems. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 159

4 Rigid Body Mechanics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1634.1 General Considerations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1634.2 Distribution of Velocities and Accelerations in a Rigid Body . . . 1654.3 Inertial Forces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 168

4.3.1 Action of the Coriolis Force on the Motion of Bodiesat the Surface of the Earth . . . . . . . . . . . . . . . . . . . . 170

4.3.2 Foucault’s Pendulum . . . . . . . . . . . . . . . . . . . . . . . . 1724.4 Euler’s Angles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1754.5 Motion of a Rigid Body About a Fixed Point . . . . . . . . . . . . . 177

4.5.1 Kinematic Preliminaries . . . . . . . . . . . . . . . . . . . . . . 1784.5.2 Angular Momentum . . . . . . . . . . . . . . . . . . . . . . . . . 1794.5.3 Kinetic Energy . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1804.5.4 Ellipsoid of Inertia . . . . . . . . . . . . . . . . . . . . . . . . . . 1824.5.5 Euler’s Equations of Motion . . . . . . . . . . . . . . . . . . . 183

4.6 Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1884.6.1 Physical Pendulum . . . . . . . . . . . . . . . . . . . . . . . . . . 1884.6.2 Symmetrical Top . . . . . . . . . . . . . . . . . . . . . . . . . . . 1894.6.3 Fast Top. The Gyroscope . . . . . . . . . . . . . . . . . . . . . 1974.6.4 Motion of a Rigid Body Relative to a Non-inertial

Frame. The Gyrocompass . . . . . . . . . . . . . . . . . . . . . 1984.6.5 Motion of Rigid Bodies in Contact . . . . . . . . . . . . . . 2024.6.6 Mechanical–Electromagnetic Analogies . . . . . . . . . . . 205

4.7 Problems. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 208

5 Hamiltonian Formalism . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2115.1 Hamilton’s Canonical Equations . . . . . . . . . . . . . . . . . . . . . . 211

5.1.1 Motion of a Particle in a Plane . . . . . . . . . . . . . . . . . 2165.1.2 Motion of a Particle Relative to a Non-inertial Frame . 2185.1.3 Motion of a Charged Particle in an

Electromagnetic Field . . . . . . . . . . . . . . . . . . . . . . . . 2195.1.4 Energy of a Magnetic Dipole in an External Field . . . . 222

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5.2 Routh’s Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2225.3 Poisson Brackets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 225

5.3.1 Poisson Brackets for Angular Momentum . . . . . . . . . . 2285.3.2 Poisson Brackets and Commutators . . . . . . . . . . . . . . 2315.3.3 Lagrange Brackets . . . . . . . . . . . . . . . . . . . . . . . . . . 234

5.4 Canonical Transformations . . . . . . . . . . . . . . . . . . . . . . . . . . 2365.4.1 Extensions and Applications . . . . . . . . . . . . . . . . . . . 2415.4.2 Mechanical–Thermodynamical Analogy.

Thermodynamic Potentials . . . . . . . . . . . . . . . . . . . . 2475.5 Infinitesimal Canonical Transformations . . . . . . . . . . . . . . . . . 249

5.5.1 Total Momentum as Generator of Translations . . . . . . 2515.5.2 Total Angular Momentum as Generator

of Rotations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2515.5.3 Hamiltonian as Generator of Time-Evolution . . . . . . . 252

5.6 Integral Invariants . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2535.6.1 Integral Invariants of the Canonical Equations. . . . . . . 2555.6.2 The Relative Universal Invariant of Mechanics . . . . . . 2565.6.3 Liouville’s Theorem . . . . . . . . . . . . . . . . . . . . . . . . . 2585.6.4 Pfaff Forms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2615.6.5 Quantum Mechanical Harmonic Oscillator . . . . . . . . . 263

5.7 Hamilton–Jacobi Formalism . . . . . . . . . . . . . . . . . . . . . . . . . 2655.7.1 Hamilton–Jacobi Equation. . . . . . . . . . . . . . . . . . . . . 2655.7.2 Methods for Solving the Hamilton–Jacobi Equation . . . 2685.7.3 Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2735.7.4 Action–Angle Variables . . . . . . . . . . . . . . . . . . . . . . 2785.7.5 Adiabatic Invariants . . . . . . . . . . . . . . . . . . . . . . . . . 287

5.8 Problems. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 289

6 Mechanics of Continuous Deformable Media . . . . . . . . . . . . . . . . 2936.1 General Considerations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2936.2 Kinematics of Continuous Deformable Media . . . . . . . . . . . . . 294

6.2.1 Lagrange’s Method . . . . . . . . . . . . . . . . . . . . . . . . . 2946.2.2 Euler’s Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . 295

6.3 Dynamics of Continuous Media . . . . . . . . . . . . . . . . . . . . . . . 2976.3.1 Equation of Continuity . . . . . . . . . . . . . . . . . . . . . . . 2976.3.2 Forces Acting upon a Continuous Deformable

Medium . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3006.3.3 General Theorems . . . . . . . . . . . . . . . . . . . . . . . . . . 3016.3.4 Equations of Motion of a CDM. Cauchy Stress

Tensor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3026.4 Deformation of a Continuous Deformable Medium About

a Point. Linear Approximation. . . . . . . . . . . . . . . . . . . . . . . . 3066.4.1 Rotation Tensor and Small-Strain Tensor . . . . . . . . . . 3066.4.2 Saint-Venant Compatibility Conditions . . . . . . . . . . . . 3096.4.3 Finite-Strain Tensor . . . . . . . . . . . . . . . . . . . . . . . . . 310

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6.5 Elastic Medium . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3116.5.1 Hooke’s Generalized Law . . . . . . . . . . . . . . . . . . . . . 3116.5.2 Equations of Motion of an Elastic Medium. . . . . . . . . 3156.5.3 Plane Waves in Isotropic Elastic Media . . . . . . . . . . . 317

6.6 Perfect Fluid . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3196.6.1 Equation of Motion of a Perfect Fluid . . . . . . . . . . . . 3196.6.2 Particular Types of Motion of an Ideal Fluid. . . . . . . . 3206.6.3 Fundamental Conservation Theorems . . . . . . . . . . . . . 3286.6.4 Magnetodynamics of Ideal Fluids. . . . . . . . . . . . . . . . 338

6.7 Viscous Fluid . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3396.8 Lagrangian Formalism . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 344

6.8.1 Euler–Lagrange Equations for Continuous Systems . . . 3446.8.2 Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 347

6.9 Hamiltonian Formalism . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3576.9.1 Hamilton’s Canonical Equations

for Continuous Systems . . . . . . . . . . . . . . . . . . . . . . 3576.9.2 Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 362

6.10 Noether’s Theorem for Continuous Systems . . . . . . . . . . . . . . 3736.10.1 Hamilton’s Principle and the Equations of Motion . . . . 3736.10.2 Symmetry Transformations . . . . . . . . . . . . . . . . . . . . 3776.10.3 Energy Conservation Theorem. . . . . . . . . . . . . . . . . . 3806.10.4 Momentum Conservation Theorem . . . . . . . . . . . . . . 3816.10.5 Angular Momentum Conservation Theorem . . . . . . . . 3826.10.6 Centre of Mass Theorem. . . . . . . . . . . . . . . . . . . . . . 383

6.11 Problems. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 385

Addenda . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 387

Addendum I: Special Theory of Relativity . . . . . . . . . . . . . . . . . . . . . 389

Addendum II: Quantum Theory and the Atom . . . . . . . . . . . . . . . . . 399

Addendum III: Stochastic Processes and the Langevin Equation . . . . . 409

Appendix A: Elements of Vector and Tensor Algebra. . . . . . . . . . . . . 411

Appendix B: Elements of Vector and Tensor Analysis . . . . . . . . . . . . 427

References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 447

Author Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 449

Subject Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 451

Contents xvii