Applications of the Theory of Groups in Mechanics and Physics / Edition 1

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We present applications of the group theory to the solution and systematization of several problems in the theory of differential equations (for instance, a base for the method of separation of variables), classical mechanics (for instance, the analytic form of the Lagrangian), relativity theory, quantum mechanics, and elementary particle physics. By this comprehensive work we wish to give the reader a number of preliminary notions and examples which are absolutely necessary for a better understanding of certain works, in different domains, which are based on applications of group theory. The present volume is a new edition of a volume published in 1985, ("Aplicatii ale teoriei grupurilor in mecanica si f zica", 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 Poincar groups of relativity theory). The work requires as preliminaries only the mathematical knowledge acquired by a student in a technical university. It is addressed to a large audience, to all those interested and compelled to use mathematical methods in various fields of research, such as mechanics, physics, engineering, people involved in research or teaching, as well as students.

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Editorial Reviews

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From the reviews:

"This book will be of interest for those readers aiming to have a view of the applications of group theory to several important questions in classical and quantum mechanics, as well as in the theory of differential equations. … The amount of topics treated is ample, and each one is developed in detail … so the book will be useful also for students. … The book contains a number of cross references between material in different chapters, making the text clearer and self-consistent." (Arturo Ramos, Mathematical Reviews, Issue 2006 h)

"In the book a many of applications of the group theory to the solution and systematization of problems in the theory of differential equations, classical mechanics, relativity theory, quantum mechanics and elementary particle physics are presented. ... The book provides a simple introduction to the subject and requires as preliminaries only the mathematical knowledge acquired by a student in a technical university." (A. A. Bogush, Zentralblatt MATH, Vol. 1072 (23), 2005)

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Product Details

  • ISBN-13: 9781402020469
  • Publisher: Springer Netherlands
  • Publication date: 4/30/2004
  • Series: Fundamental Theories of Physics Series, #140
  • Edition description: 2004
  • Edition number: 1
  • Pages: 446
  • Product dimensions: 6.14 (w) x 9.21 (h) x 1.00 (d)

Table of Contents

1. Elements of General Theory of Groups.- 1 Basic notions.- 1.1 Introduction of the notion of group.- 1.2 Basic definitions and theorems.- 1.3 Representations of groups.- 1.4 The S3 group.- 2 Topological groups.- 2.1 Definitions. Generalities. Lie groups.- 2.2 Lie algebras. Unitary representations.- 3 Particular Abelian groups.- 3.1 The group of real numbers.- 3.2 The group of discrete translations.- 3.3 The SO(2) and Cn, groups.- 2. Lie Groups.- 1 The SO(3) group.- 1.1 Rotations.- 1.2 Parametrization of SO(3) and O(3).- 1.3 Functions defined on O(3). Infinitesimal generators.- 2 The SU(2) group.- 2.1 Parametrization of SU(2).- 2.2 Functions defined on SU(2). Infinitesimal generators.- 3 The SU(3) and GL(n,—) groups.- 3.1 SU(3) Lie algebra.- 3.2 Infinitesimal generators. Parametrization of SU(3).- 3.3 The GL(n,—) and SU(n) groups.- 4 The Lorentz group.- 4.1 Lorentz transformations.- 4.2 Parametrization and infinitesimal generators.- 3. Symmetry Groups of Differential Equations.- 1 Differential operators.- 1.1 The SO(3) and SO(n) groups.- 1.2 The SU(2) and SU(3) groups.- 2 Invariants and differential equations.- 2.1 Preliminary considerations.- 2.2 Invariant differential operators.- 3 Symmetry groups of certain differential equations.- 3.1 Central functions. Characters.- 3.2 The SO(3), SU(2), and SU(3) groups.- 3.3 Direct products of irreducible representations.- 4 Methods of study of certain differential equations.- 4.1 Ordinary differential equations.- 4.2 The linear equivalence method.- 4.3 Partial differential equations.- 4. Applications in Mechanics.- 1 Classical models of mechanics.- 1.1 Lagrangian formulation of classical mechanics.- 1.2 Hamiltonian formulation of classical mechanics.- 1.3 Invariance of the Lagrange and Hamilton equations.- 1.4 Noether’s theorem and its reciprocal.- 2 Symmetry laws and applications.- 2.1 Lie groups with one parameter and with m parameters.- 2.2 The Symplectic and Euclidean groups.- 3 Space-time symmetries. Conservation laws.- 3.1 Particular groups. Noether’s theorem.- 3.2 The reciprocal of Noether’s theorem.- 3.3 The Hamilton-Jacobi equation for a free particle.- 4 Applications in the theory of vibrations.- 4.1 General considerations.- 4.2 Transformations of normal coordinates.- 5. Applications in the Theory of Relativity and Theory of Classical Fields.- 1 Theory of Special Relativity.- 1.1 Preliminary considerations.- 1.2 Applications in the theory of Special Relativity.- 2 Theory of electromagnetic field.- 2.1 Noether’s theorem for the electromagnetic field.- 2.2 Conformal transformations in four dimensions.- 3 Theory of gravitational field.- 3.1 General equations.- 3.2 Conservation laws in the Riemann space.- 6. Applications in Quantum Mechanics and Physics of Elementary Particles.- 1 Non-relativistic quantum mechanics.- 1.1 Invariance properties of quantum systems.- 1.2 The angular momentum. The spin.- 2 Internal symmetries of elementary particles.- 2.1 The isospin and the SU(2) group.- 2.2 The unitary spin and the SU(3) group.- 3 Relativistic quantum mechanics.- 3.1 Basic equations. Symmetry groups.- 3.2 Elementary particle interactions.- References.

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