Applications Of The Theory Of Groups In Mechanics And Physics

by Teodorescu, Petre P.; Nicorovici, Nicolae A.

• ISBN13: 9781402020469
• ISBN10: 1402020465
• Format: Hardcover
• Copyright: 2004-04-30
• Publisher: Kluwer Academic Pub

Summary

The present volume is a new edition of a volume published in 1985, ("Aplicatii ale teoriei grupurilor in mecanica si fízica", Editura Tehnica, Bucharest, Romania). This new edition contains many improvements concerning the presentation, as well as new topics using an enlarged and updated bibliography. In addition to the large area of domains in physics covered by this volume, we are presenting both discrete and continuous groups, while most of the books about applications of group theory in physics present only one type of groups (i.e., discrete or continuous), and the number of analyzed groups is also relatively small (i.e., point groups of crystallography, or the groups of rotations and translations as examples of continuous groups; some very specialized books study the Lorentz and Poincaré groups of relativity theory).

Table of Contents

PREFACE	ix
INTRODUCTION	xi
1. ELEMENTS OF GENERAL THEORY OF GROUPS	1	(60)
1 Basic notions	1	(39)
1.1 Introduction of the notion of group	1	(3)
1.2 Basic definitions and theorems	4	(15)
1.3 Representations of groups	19	(14)
1.4 The S3 group	33	(7)
2 Topological groups	40	(12)
2.1 Definitions. Generalities. Lie groups	40	(10)
2.2 Lie algebras. Unitary representations	50	(2)
3 Particular Abelian groups	52	(9)
3.1 The group of real numbers	52	(2)
3.2 The group of discrete translations	54	(4)
3.3 The SO(2) and Cn, groups	58	(3)
2. LIE GROUPS	61	(62)
1 The SO(3) group	61	(15)
1.1 Rotations	61	(9)
1.2 Parametrization of SO(3) and 0(3)	70	(4)
1.3 Functions defined on 0(3). Infinitesimal generators	74	(2)
2 The SU(2) group	76	(13)
2.1 Parametrization of SU(2)	76	(6)
2.2 Functions defined on SU(2). Infinitesimal generators	82	(7)
3 The SU(3) and GL(n,C) groups	89	(22)
3.1 SU(3) Lie algebra	89	(13)
3.2 Infinitesimal generators. Parametrization of SU(3)	102	(5)
3.3 The GL(n,C) and SU(n) groups	107	(4)
4 The Lorentz group	111	(12)
4.1 Lorentz transformations	111	(7)
4.2 Parametrization and infinitesimal generators	118	(5)
3. SYMMETRY GROUPS OF DIFFERENTIAL EQUATIONS	123	(78)
1 Differential operators	123	(10)
1.1 The SO(3) and SO(n) groups	123	(4)
1.2 The SU(2) and SU(3) groups	127	(6)
2 Invariants and differential equations	133	(16)
2.1 Preliminary considerations	133	(5)
2.2 Invariant differential operators	138	(11)
3 Symmetry groups of certain differential equations	149	(27)
3.1 Central functions. Characters	149	(2)
3.2 The SO(3), SU(2), and SU(3) groups	151	(8)
3.3 Direct products of irreducible representations	159	(17)
4 Methods of study of certain differential equations	176	(25)
4.1 Ordinary differential equations	176	(1)
4.2 The linear equivalence method	177	(9)
4.3 Partial differential equations	186	(15)
4. APPLICATIONS IN MECHANICS	201	(78)
1 Classical models of mechanics	201	(29)
1.1 Lagrangian formulation of classical mechanics	201	(5)
1.2 Hamiltonian formulation of classical mechanics	206	(7)
1.3 Invariance of the Lagrange and Hamilton equations	213	(7)
1.4 Noether's theorem and its reciprocal	220	(10)
2 Symmetry laws and applications	230	(14)
2.1 Lie groups with one parameter and with m parameters	230	(5)
2.2 The Symplectic and Euclidean groups	235	(9)
3 Space-time symmetries. Conservation laws	244	(19)
3.1 Particular groups. Noether's theorem	244	(7)
3.2 The reciprocal of Noether's theorem	251	(8)
3.3 The Hamilton-Jacobi equation for a free particle	259	(4)
4 Applications in the theory of vibrations	263	(16)
4.1 General considerations	263	(2)
4.2 Transformations of normal coordinates	265	(14)
5. APPLICATIONS IN THE THEORY OF RELATIVITY AND THEORY OF CLASSICAL FIELDS	279	(56)
1 Theory of Special Relativity	279	(23)
1.1 Preliminary considerations	279	(10)
1.2 Applications in the theory of Special Relativity	289	(13)
2 Theory of electromagnetic field	302	(22)
2.1 Noether's theorem for the electromagnetic field	302	(15)
2.2 Conformal transformations in four dimensions	317	(7)
3 Theory of gravitational field	324	(11)
3.1 General equations	324	(5)
3.2 Conservation laws in the Riemann space	329	(6)
6. APPLICATIONS IN QUANTUM MECHANICS AND PHYSICS OF ELEMENTARY PARTICLES	335	(88)
1 Non-relativistic quantum mechanics	335	(36)
1.1 Invariance properties of quantum systems	335	(23)
1.2 The angular momentum. The spin	358	(13)
2 Internal symmetries of elementary particles	371	(25)
2.1 The isospin and the SU(2) group	371	(12)
2.2 The unitary spin and the SU(3) group	383	(13)
3 Relativistic quantum mechanics	396	(27)
3.1 Basic equations. Symmetry groups	396	(11)
3.2 Elementary particle interactions	407	(16)
REFERENCES	423	(8)
INDEX	431