ENCYCLOPEDIA OF PHYSICS

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1 ENCYCLOPEDIA OF PHYSICS EDITED BY S. FLOGGE VOLUME 111/1 PRINCIPLES OF CLASSICAL MECHANICS AND FIELD THEORY WITH 106 FIGURES SPRINGER-VERLAG BERLIN HEIDELBERG GMBH 1960

2 HANDBUCH DER PHYSIK HERAUSGEGEBEN VON S. FLOGGE BAND III/1 PRINZIPIEN DER KLASSISCHEN MECHANIK UNO FELDTHEORIE MIT 106 FIGUREN SPRINGER-VERLAG BERLIN HEIDELBERG GMBH 1960

3 ISBN ISBN (ebook) DOI / Alle Rechte, insbesondere das der Obersetzung in fremde Sprachen, vorbehalten. Ohne ausdriickliche Genehmigung des Verlages ist es auch nicht gestattet, dieses Buch oder Teile daraus auf photomechanischem Wege (Photokopie, Mikrokopie) zu vervielfiiltigen. by Springer-Verlag Berlin Heidelberg 1960 Urspriinglich erschienen bei Springer-Verlag OHG. Berlin Giittingen Heidelberg 1960 Softcover reprint oftbe hardcover 1st edition 1960 Die Wiedergabe von Gebrauchsnamen, Handelsnamen, Warenbezeichnungen usw. in diesem Werk berechtigt auch ohne besondere Kennzeichnung nicht zu der Annahme, dab solche Namen im Sinn der Warenzeichen- und Markenschutz Gesetzgebung als frei zu betrachten waren und daher von jedermann benutzt werden diirften.

4 Contents. Classical Dynamics. By Professor Dr. JOHN L. SYNGE, School of Theoretical Physics, Institute for Advanced Studies, Dublin (Ireland). (With 57 Figures) A. Introduction B. Kinematics. 12 I. Displacements of rigid bodies 12 II. Kinematics. 25 III. Mass distributions and force systems. 31 IV. Generalized coordinates 38 C. Dynamics of a particle. 43 I. Equations of motion. 43 II. One-dimensional motions 46 III. Two-dimensional motions. 48 IV. Three-dimensional motions 53 D. Dynamics of systems of particles and of rigid bodies. 56 I. Equations of motion. 56 II. Systems without constraints 69 III. Rigid body with a fixed point. 82 E. General dynamical theory 98 I. Geometrical representations of dynamics 98 II. The space of events(q T). 105 III. Momentum-energy space (PH) 130 IV. Configuration space (Q) v. The space of states and energy ( Q T PH) 143 VI. The space of states ( Q T P) 163 VII. Phase space (PQ) 167 VIII. Small oscillations 180 F. Relativistic dynamics 198 I. Minkowskian space-time and the laws of dynamics. 198 II. Some special dynamical problems 210 III. De Broglie waves 215 IV. Relativistic catastrophes 217 General references. 223 Page The Classical Field Theories. By Professor Dr. C. TRUESDELL, Bloomington, Indiana and Dr. R.A. TouPIN, U.S. Naval Research Laboratory, Washington, D.C. (USA). (With 47 Figures) A. The field viewpoint in classical physics. 226 B. Motion and mass I. Deformation a) Deformation gradients. 241 b) Strain c) Rotation d) Special deformations 283 e) Small deformation 303 f) Oriented bodies

5 VI Contents. II. Motion a) Velocity b) Material systems 337 c) Stretching and spin 347 d) Acceleration e) Special developments concerning vorticity 385 ei) The vorticity field ell) Vorticity averages e III) Bernoullian theorems eiv) Convection and diffusion of vorticity. 409 f) Further special motions 430 g) Relative motion 437 III. Mass a) Definition of mass b) Solution of the equation of continuity 474 c) Momentum C. Singular surfaces and waves I. Geometry of singular surfaces. 492 II. The motion of surfaces III. Kinematics of singular surfaces 503 IV. Singular surfaces associated with a motion 506 V. Discontinuous equations of balance 525 D. Stress I. The balance of momentum II. The stress principle III. Applications of CAUCHY's laws 568 IV. General solutions of the equations of motion 582 V. Variational principles 594 E. Energy and entropy I. The balance of energy 608 II. Entropy a) The caloric equation of state 615 b) The production of entropy 638 III. Equilibrium F. Charge and magnetic flux 660 I. Introduction II. The conservation of charge and magnetic flux. 666 III. The Maxwell-Lorentz aether relations 677 IV. Conservation of energy and momentum. 689 G. Constitutive equations I. Generalities II. Examples of kinematical constitutive equations 704 III. Example of an energetic constitutive equation. 709 IV. Examples of mechanical constitutive equations 710 V. Examples of thermo-mechanical constitutive equations. 734 VI. Electromagnetic constitutive equations VII. Electromechanical constitutive equations List of works cited Additional Bibliography K: Kinematics of special motions (geometrical theory) 784 Additional Bibliography N: Non-relativistic kinematics and mechanics in generalized spaces.... Additional Bibliography P: Principles of mechanics.... Additional Bibliography R: Relativistic continuum theories

6 Contents. VII Appendix. Tensor Fields. By Dr. J. L. ERICKSEN, Associate Professor of Theoretical Mechanics, Mechanical Engineering Department, Johns Hopkins University, Baltimore, Md. (USA). (With 2 Figures) I. Preliminaries a) Notation b) Use of complex co-ordinates II. Dimensions and physical components. 797 a) Dimensions of a tensor and its components. 797 b) Physical components III. Double tensor fields a) Definition and examples. 805 b) The total covariant derivative 810 IV. Integrals of tensor fields 813 a) Preliminaries b) Circulation, flux, total, and moments 814 c) The transformation of GREEN and KELVIN 815 V. Vector fields a) Vector lines, sheets, and tubes b) Special classes of fields c) Potentials VI. Tensors of order two..... a) Proper numbers and vectors b) Powers and matrix polynomials. c) Decompositions d) Normal and shear components e) Tensor lines and sheets References.... Sachverzeichnis (Deutsch-Englisch) Subject Index (English-German)

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