Structure of Dynamical Systems: A Symplectic View of Physics (Progress in Mathematics)

J.m. Souriau

ISBN 10: 1461266920 / ISBN 13: 9781461266921

Editore: Birkhaeuser, 2012

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reprint edition. 444 pages. 9.25x6.10x1.00 inches. In Stock. Codice articolo __1461266920

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Titolo: Structure of Dynamical Systems: A Symplectic...

Casa editrice: Birkhaeuser

Data di pubblicazione: 2012

Legatura: Paperback

Condizione: Brand New

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Riassunto:

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. 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.

Contenuti:

I. 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 gases.- Equilibria of a conservative system.- Ideal gases.- A monatomic ideal gas.- An arbitrary ideal gas.- An ideal gas thermometer.- Heat and work.- Specific heat.- Covariant statistical mechanics.- Examples.- The statistical equilibrium of an isolated system.- Relativistic statistical mechanics.- A relativistic ideal gas.- Statistical equilibria of photons.- V. A Method of Quantization.- §18. Geometric quantization.- Prequantum manifolds.- Prequantization of a symplectic manifold.- Prequantization of a symplectic manifold admitting a potential.- Prequantization of a sphere S2.- Prequantization by “fusion”.- Prequantization of a direct product.- Prequantization of a relativistic particle with spin ½.- Prequantization of a massless particle.- Massless particle with spin ½.- Massless particle with spin 1.- Planck manifolds.- Quantomorphisms.- Homotopy and prequantization.- Systems of elementary particles.- Infinitesimal quantomorphisms.- Quantization of dynamical groups.- The Hilbert space of a prequantum manifold.- §19. Quantization of dynamical systems.- The correspondence principle.- State vectors and observables.- The formulation of Planck’s condition.- Stationary states.- The formation of wave equations.- The nonrelativistic material point.- The relativistic material point.- The nonrelativistic particle with spin ½.- The relativistic particle with spin ½.- The massless particle with spin ½.- The massless particle with spin 1.- Assemblées of particles.- Creation and annihilation operators.- Quantum states.- List of notation.