Theoretical Mechanics: Theoretical Physics 1

by Reiner M. Dreizler, Cora S. Lüdde

Overview

This book is the first of a series covering the major topics that are taught in university courses in Theoretical Physics: Mechanics, Electrodynamics, Quantum Theory and Statistical Physics. After an introduction to basic concepts of mechanics more advanced topics build the major part of this book. Interspersed is a discussion of selected problems of motion. This is followed by a concise treatment of the Lagrangian and the Hamiltonian formulation of mechanics, as well as a brief excursion on chaotic motion. The last chapter deals with applications of the Lagrangian formulation to specific systems (coupled oscillators, rotating coordinate systems, rigid bodies). The level of the last sections is advanced. The text is accompanied by an extensive collection of online material, in which the possibilities of the electronic medium are fully exploited, e.g. in the form of applets, 2D- and 3D-animations. It contains: A collection of 74 problems with detailed step-by-step guidance towards the solutions, a collection of comments and additional mathematical details in support of the main text, a complete presentation of all the mathematical tools needed.

Product Details

ISBN-13:	9783642111372
Publisher:	Springer Berlin Heidelberg
Publication date:	11/05/2010
Series:	Graduate Texts in Physics
Edition description:	2010
Pages:	402
Product dimensions:	6.40(w) x 9.30(h) x 1.30(d)

About the Author

Reiner Dreizler, professor emeritus of Theoretical Physics at the JWG University, Frankfurt am Main. Studies of physics and mathematics in Freiburg (Dipl. Phys.) and Canberra (Ph.D.). Appointments as Research Associate and Assistant Professor at the University of Pennsylvania in Philadelphia, since 1972 Full Professor in Frankfurt. Honours achieved are Fellow of the American Physical Society and the endowed S. Lyson Professorship. The scientific work covers many body aspects in nuclear, atomic, molecular, solid state and elementary particle physics, both for structural as well as collision problems. The publication list (together with co-workers from around the globe) includes approx. 250 scientific papers, 5 conference proceedings and 4 text books. A teaching experience of more than 40 years in all fields of Theoretical Physics and related mathematical disciplines is the basis for the forthcoming introductory series in Theoretical Physics. Cora Lüdde, studies of physics and mathematics at the JWG University in Frankfurt am Main (Dipl.Phys.). A break from scientific activities in order to educate a daughter was followed by intensive occupation with computational matters, in particular the design of web based applications. Extensive programming experience in word processing and visualisation languages as e.g. Latex, Flash, Perl, Java, Fortran … on Windows and Unix platforms. Co-author of 3 books with supporting CD-ROMs on Theoretical Physics at university level. Studies of pedagogical matters dealing in particular with the teaching of sience at all levels of school and university.

Table of Contents

1 A First Survey 1

2 Kinematics 13

2.1 One-dimensional motion 13

2.1.1 Three examples for the motion in one space dimension 14

2.1.2 Velocity 19

2.1.3 Acceleration 22

2.1.4 First remarks concerning dynamical aspects 25

2.2 Problems of motion in two or three dimensions 27

2.2.1 Two-dimensional motion 29

2.2.2 Motion in three spatial dimensions 36

2.2.3 An example for the determination of trajectories in two space dimensions 38

2.3 Vectorial formulation of problems of motion 40

2.3.1 Basic concepts 41

2.3.2 Vectorial description of motion 43

2.3.3 Area theorem 48

2.4 Curvilinear coordinates 53

2.4.1 Coordinates in the plane 53

2.4.2 Spatial coordinates 60

3 Dynamics I: Axioms and Conservation Laws 67

3.1 The axioms of mechanics 67

3.1.1 The concept of force 67

3.1.2 Inertial and gravitational mass 69

3.1.3 The axioms 72

3.1.4 The first axiom: inertial systems 72

3.1.5 The second axiom: momentum 76

3.1.6 The third axiom: interactions 77

3.2 The conservation laws of mechanics 84

3.2.1 The momentum principle and momentum conservation 84

3.2.2 The angular momentum principle and angular momentum conservation 91

3.2.3 Energy and energy conservation for a mass point 102

3.2.4 Energy conservation for a system of mass points 122

3.2.5 Application: collision problems 130

4 Dynamics II: Problems of Motion 139

4.1 Kepler's problem 139

4.1.1 Preliminaries 140

4.1.2 Planetary motion 141

4.1.3 Comets and meteorites 155

4.2 Oscillator problems 160

4.2.1 The mathematical pendulum 161

4.2.2 The damped harmonic oscillator 169

4.2.3 Forced oscillations: harmonic restoring forces 173

4.2.4 Forced oscillations: general excitations 180

5 General Formulation of the Mechanics of Point Particles 185

5.1 Lagrange I: the Lagrange equations of the first kind 186

5.1.1 Examples for the motion under constraints 186

5.1.2 Lagrange I for one point particle 192

5.2 D'Alembert's principle 204

5.2.1 D'Alembert's principle for one mass point 204

5.2.2 D'Alembert's principle for systems of point particles 209

5.3 The Lagrange equations of the second kind (Lagrange II) 214

5.3.1 Lagrange II for one point particle 214

5.3.2 Lagrange II and conservation laws for one point particle 231

5.3.3 Lagrange II for a system of mass points 241

5.4 Hamilton's formulation of mechanics 246

5.4.1 Hamiltion's principle 246

5.4.2 Hamiltion's equation of motion 254

5.4.3 A cursory look into phase space 262

6 Application of the Lagrange Formalism 271

6.1 Coupled harmonic oscillators 271

6.1.1 Coupled oscillating system: two masses and three springs 272

6.1.2 Beats 276

6.1.3 The linear oscillator chain 278

6.1.4 The differential equation of an oscillating string 290

6.2 Rotating coordinate systems 294

6.2.1 Simple manifestation of apparent forces 295

6.2.2 General discussion of apparent forces in rotating coordinate systems 297

6.2.3 Apparent forces and the rotating earth 305

6.3 The motion of rigid bodies 314

6.3.1 Preliminaries 315

6.3.2 The kinetic energy of rigid bodies 316

6.3.3 The structure of the inertia matrix 322

6.3.4 The angular momentum of a rigid body 332

6.3.5 The Euler angles 334

6.3.6 The equations of motion for the rotation of a rigid body 337

6.3.7 Rotational motion of rigid bodies 340

References 353

Appendix 357

A Biographical data 359

B The Greek Alphabet 365

C Nomenclature 367

D Physical Quantities 369

E Some Constants and Astronomical Data 373

F Formulae 377

F.1 Plane Polar Coordinates 377

F.2 Cylinder Coordinates 378

F.3 Spherical Coordinates 378

F.4 Sum Formulae / Moivre Formula 379

F.5 Hyperbolic Functions 380

F.6 Series Expansions 380

F.7 Approximations (δ small) 380

G Problems on the virtual CD-ROM 381

Index 397