Geometry from Dynamics, Classical and Quantum (Hardcover)

Foreword: The birth and the long gestation of a project.- Some examples of linear and nonlinear physical systems and their dynamical equations.- Equations of the motion for evolution systems.- Linear systems with infinite degrees of freedom.- Constructing nonlinear systems out of linear ones.- The language of geometry and dynamical systems: the linearity paradigm.- Linear dynamical systems: The algebraic viewpoint.- From linear dynamical systems to vector fields.- Exterior differential calculus on linear spaces.- Exterior differential calculus on submanifolds.- A tensorial characterization of linear structures.- Partial linear structures: Vector bundles.- Covariant calculus.- Riemannian and Pseudo-Riemannian metrics on linear vector spaces.- Invariant geometric structures and the classical formulations of dynamics of Poisson, Jacobi, Hamilton and Lagrange.- Linear vector fields.- Additional invariant structures for linear vector fields.- Poisson structures.- The inverse problem for Poisson structures.- Symplectic structures.- Lagrangian structures.- Invariant Hermitean structures and the geometry of quantum systems.- Invariant Hermitean inner products.- Complex structures and complex exterior calculus.- Algebras associated with Hermitean structures.- The geometry of quantum dynamical evolution.- The Geometry of Quantum Mechanics and the GNS construction.- Alternative Hermitean structures for quantum systems.- Folding and unfolding Classical and Quantum systems.- Introduction: separable dynamics.- The geometrical description of reduction.- The algebraic description.- Reduction in Quantum Mechanics.- Integrable and superintegrable systems.- The geometrization of the notion of integrability.- The normal form of an integrable system.- Lax representation.- The Calogero system: inverse scattering.- Lie-Scheffers systems.- The inhomogeneous linear equation revisited.- Inhomogeneous linear systems.- Non-linear superposition rule.- Related maps.- Lie systems on Lie groups and homogeneous spaces.- Some examples of Lie systems.- Hamiltonian systems of Lie type.