Detailansicht

InhaltsangabeI. Differential Geometry.- §1. Manifolds.- The definition of a manifold.- Open sets.- Differentiable maps.- The tangent space.- Submanifolds.- Manifolds defined by an equation.- Covering spaces.- Quotient manifolds.- Connectedness.- Homotopy.- §2. Derivations.- Variables.- Vector fields and derivations.- Derivations of linear operators.- The image of a vector field.- Lie brackets.- §3. Differential equations.- The exponential of a vector field.- The image of a differential equation.- The derivative of the exponential map.- §4. Differential forms.- Covariant fields.- The inverse image of a covariant field.- The Lie derivative.- Covariant tensor fields.- p- Forms.- The exterior derivative.- §5. Foliated manifolds.- Foliations.- The quotient of a manifold by a foliation.- Integral invariants.- The characteristic foliation of a form.- §6. Lie groups.- Actions of a Lie group on a manifold.- The Lie algebra of a Lie group.- Orbits.- The adjoint representation.- Lie subalgebras and Lie subgroups.- The stabilizer.- Classical examples of Lie groups.- Euclidean spaces.- Matrix realizations.- §7. The calculus of variations.- Classical variational problems.- Canonical variables.- The Hamiltonian formalism.- A geometrical interpretation of the canonical equations.- Transformations of a variational problem.- Noether's theorem.- II. Symplectic Geometry.- §8. 2-Forms.- Orthogonality.- Canonical bases.- The symplectic group.- §9. Symplectic manifolds.- Symplectic and presymplectic manifolds.- Symplectic structures arising from a 1-form.- Poisson brackets.- Induced symplectic structures.- §10. Canonical transformations.- Canonical charts.- Canonical transformations.- Canonical similitudes.- Covering spaces of symplectic manifolds.- Infinitesimal canonical transformations.- §11. Dynamical Groups.- The definition of a dynamical group.- The cohomology of a dynamical group.- The cohomology of a Lie group.- The cohomology of a Lie algebra.- Symplectic manifolds defined by a Lie group.- III. Mechanics.- §12. The geometric structure of classical mechanics.- Material points.- Systems of material points.- Constraints.- Describing forces.- The evolution space.- Phase spaces and the space of motions.- The Lagrange 2-form.- The Lagrange form for constrained systems.- Changing the reference frame.- The principle of Galilean relativity.- Maxwell's principle.- Potentials and the variational formalism.- Geometric consequences of Maxwell's principle.- An application: variation of constants.- Galilean moments.- Remarks.- Examples of dynamical groups.- §13. The principles of symplectic mechanics.- Nonrelativistic symplectic mechanics.- Moments, mass, and the center of mass.- The center of mass decomposition.- Minkowski space and the Poincaré group.- Relativistic mechanics.- §14. A mechanistic description of elementary particles.- Elementary systems.- A particle with spin.- Remarks.- A particle without spin.- A massless particle.- Remarks.- Nonrelativistic particles.- Mass and barycenter of a relativistic system.- Inversions of space and time.- A particle with nonzero mass.- A massless particle.- §15. Particle dynamics.- A material point in an electromagnetic field.- A particle with spin in an electromagnetic field.- Systems of particles without interactions.- Interactions.- Scattering theory.- Bounded scattering sources.- Geometrical optics.- Planar mirrors.- Collisions of free particles.- IV. Statistical Mechanics.- §16. Measures on a manifold.- Composite manifolds.- Compact sets.- Riesz spaces.- Measures.- The tensor product of measures.- Examples of measures.- Completely continuous measures.- Examples of completely continuous measures.- The support of a measure.- Bounded measures.- Integrable functions.- The image of a measure.- Examples.- Random variables.- Average values.- Entropy and Gibbs measures.- The Gibbs canonical ensemble of a dynamical group.- §17. The principles of statistical mechanics.- Statistical states.- Hypotheses of the kinetic theory of gas

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.