Structure Of Dynamical Systems
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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) quantization.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.
Introduction1Manifolds2Derivations3Differential equations4Differential forms5Foliated manifolds6Lie groups7The calculus of variations82-Forms9Symplectic manifolds10Canonical transformations11Dynamical Groups12The geometric structure of classical mechanics13The principles of symplectic mechanics14A mechanistic description of elementary particles15Particle dynamics16Measures on a manifold17The principles of statistical mechanics18Geometric quantization19Quantization of dynamical systemsBibliographyIndexList of notation
