Structure of Dynamical Systems

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.