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Structure of Dynamical Systems : A Symplectic View of Physics

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概要 This book is addressed to graduate students and researchers in mathematics and physics who are interested in mathematical and theoretical physics, symplectic geometry, mechanics, and (geometric) quant...ization. The aim of the book is to treat all three basic theories of physics, namely, classical mechanics, statistical mechanics, and quantum mechanics from the same perspective, that of symplectic geometry, thus showing the unifying power of the symplectic geometric approach. Reading this book will give the reader a deep understanding of the interrelationships between the three basic theories of physics. The first tow chapters provide the necessary mathematical background in differential geometry, Lie groups, and symplectic geometry. In Chapter 3 a coherent symplectic description of Galilean and relativistic mechanics is given, culminating in the classification of elementary particles (relativistic and non-relativistic, with or without spin, with or without mass). In Chapter 4 statistical mechanics is put into symplectic form, finishing with a symplectic description of the kinetic theory of gases and the computation of specific heats. Finally, in Chapter 5 the author presents his theory of geometric quantization. Highlights of this chapter are the derivations of various wave equations and the construction of the Fock space.続きを見る
目次 I. Differential Geometry
{sect}1. Manifolds
{sect}2. Derivations
{sect}3. Differential equations
{sect}4. Differential forms
{sect}5. Foliated manifolds
{sect}6. Lie groups
{sect}7. The calculus of variations
II. Symplectic Geometry
{sect}8. 2-Forms
{sect}9. Symplectic manifolds
{sect}10. Canonical transformations
{sect}11. Dynamical Groups
III. Mechanics
{sect}12. The geometric structure of classical mechanics
{sect}13. The principles of symplectic mechanics
{sect}14. A mechanistic description of elementary particles
{sect}15. Particle dynamics
IV. Statistical Mechanics
{sect}16. Measures on a manifold
{sect}17. The principles of statistical mechanics
V. A Method of Quantization
{sect}18. Geometric quantization
{sect}19. Quantization of dynamical systems
List of notation.
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登録日 2020.06.27
更新日 2020.06.28