Applications of the Theory of Groups in Mechanics and Physics / Edition 1by Petre P. Teodorescu, Nicolae-A.P. Nicorovici
Pub. Date: 12/23/2010
Publisher: Springer Netherlands
The notion of group is fundamental in our days, not only in mathematics, but also in classical mechanics, electromagnetism, theory of relativity, quantum mechanics, theory of elementary particles, etc. This notion has developed during a century and this development is connected with the names of great mathematicians as E. Galois, A. L. Cauchy, C. F. Gauss, W. R.… See more details below
The notion of group is fundamental in our days, not only in mathematics, but also in classical mechanics, electromagnetism, theory of relativity, quantum mechanics, theory of elementary particles, etc. This notion has developed during a century and this development is connected with the names of great mathematicians as E. Galois, A. L. Cauchy, C. F. Gauss, W. R. Hamilton, C. Jordan, S. Lie, E. Cartan, H. Weyl, E. Wigner, and of many others. In mathematics, as in other sciences, the simple and fertile ideas make their way with difficulty and slowly; however, this long history would have been of a minor interest, had the notion of group remained connected only with rather restricted domains of mathematics, those in which it occurred at the beginning. But at present, groups have invaded almost all mathematical disciplines, mechanics, the largest part of physics, of chemistry, etc. We may say, without exaggeration, that this is the most important idea that occurred in mathematics since the invention of infinitesimal calculus; indeed, the notion of group expresses, in a precise and operational form, the vague and universal ideas of regularity and symmetry. The notion of group led to a profound understanding of the character of the laws which govern natural phenomena, permitting to formulate new laws, correcting certain inadequate formulations and providing unitary and non contradictory formulations for the investigated phenomena.
Table of Contents1. 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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