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The book is a revised and updated version of the lectures given by the author at the University of Timi§oara during the academic year 1990-1991. Its goal is to present in detail someold and new aspects ofthe geometry ofsymplectic and Poisson manifolds and to point out some of their applications in Hamiltonian mechanics and geometric quantization. The material is organized as follows. In Chapter 1 we collect some general facts about symplectic vector spaces, symplectic manifolds and symplectic reduction. Chapter 2 deals with the study ofHamiltonian mechanics. We present here the gen eral theory ofHamiltonian mechanicalsystems, the theory ofthe corresponding Pois son bracket and also some examples ofinfinite-dimensional Hamiltonian mechanical systems. Chapter 3 starts with some standard facts concerning the theory of Lie groups and Lie algebras and then continues with the theory ofmomentum mappings and the Marsden-Weinstein reduction. The theory of Hamilton-Poisson mechan ical systems makes the object of Chapter 4. Chapter 5 js dedicated to the study of the stability of the equilibrium solutions of the Hamiltonian and the Hamilton Poisson mechanical systems. We present here some of the remarcable results due to Holm, Marsden, Ra~iu and Weinstein. Next, Chapter 6 and 7 are devoted to the theory of geometric quantization where we try to solve, in a geometrical way, the so called Dirac problem from quantum mechanics. We follow here the construc tion given by Kostant and Souriau around 1964.
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Contenuti
Introduction. Background Notations. 1. Symplectic Geometry. 2. Hamiltonian Mechanics. 3. Lie Groups; Momentum Mappings; Reduction. 4. Hamilton--Poisson Mechanics. 5. Hamiltonian Mechanical Systems and Stability. 6. Geometric Prequantization. 7. Geometric Quantization. 8. Foliated Cohomology and Geometric Quantization. 9. Symplectic Reduction; Geometric Quantization; Constrained Mechanical Systems. 10. Poisson Manifolds and Geometric Prequantization. References. Index.
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Book by Puta Mircea
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Hamiltonian Mechanical Systems and Geometric Quantization
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Hamiltonian Mechanical Systems and Geometric Quantization (Hardcover)
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Kluwer Academic Publishers, Dordrecht, 1993
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Hardcover. Condizione: new. Hardcover. This volume presents various aspects of the geometry of symplectic and Poisson manifolds, and applications in Hamiltonian mechanics and geometric quantization are indicated. Chapter 1 presents some general facts about symplectic vector space, symplectic manifolds and symplectic reduction. Chapter 2 deals with the study of Hamiltonian mechanics. Chapter 3 considers some standard facts concerning Lie groups and algebras which lead to the theory of momentum mappings and the Marsden Weinstein reduction. Chapters 4 and 5 consider the theory and the stability of equilibrium solutions of Hamilton-Poisson mechanical systems. Chapters 6 and 7 are devoted to the theory of geometric quantization. This leads, in Chapter 8, to topics such as foliated cohomology, the theory of the Dolbeault-Kostant complex, and their applications, A discussion of the relation between geometric quantization and the Marsden Weinstein reduction is presented in Chapter 9. The final chapter considers extending the theory of geometric quantization to Poisson manifolds, via the theory of symplectic groupoids.Each chapter concludes with problems and solutions, many of which present significant applications and, in some cases, major theorems. The text is suitable for graduate students and researchers whose interests and work involve symplectic geometry and Hamiltonian mechanics. Presents various aspects of the geometry of symplectic and Poisson manifolds, and applications in Hamiltonian mechanics and geometric quantization . Each chapter concludes with problems and solutions, many of which present significant applications, and in some cases, major theorems. Shipping may be from multiple locations in the US or from the UK, depending on stock availability. Codice articolo 9780792323068
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Hamiltonian Mechanical Systems and Geometric Quantization (Mathematics and Its Applications, 260)
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Hamiltonian Mechanical Systems and Geometric Quantization (Mathematics and Its Applications, 260)
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Gebunden. Condizione: New. Dieser Artikel ist ein Print on Demand Artikel und wird nach Ihrer Bestellung fuer Sie gedruckt. Introduction. Background Notations. 1. Symplectic Geometry. 2. Hamiltonian Mechanics. 3. Lie Groups Momentum Mappings Reduction. 4. Hamilton--Poisson Mechanics. 5. Hamiltonian Mechanical Systems and Stability. 6. Geometric Prequantization. 7. Geometri. Codice articolo 5966813
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