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Foreword; Acknowledgments; Contents; 1 Some Examples of Linear and Nonlinear Physical Systems and Their Dynamical Equations; 1.1 Introduction; 1.2 Equations of Motion for Evolution Systems; 1.2.1 Histories, Evolution and Differential Equations; 1.2.2 The Isotropic Harmonic Oscillator; 1.2.3 Inhomogeneous or Affine Equations; 1.2.4 A Free Falling Body in a Constant Force Field; 1.2.5 Charged Particles in Uniform and Stationary Electric and Magnetic Fields; 1.2.6 Symmetries and Constants of Motion; 1.2.7 The Non-isotropic Harmonic Oscillator
1.2.8 Lagrangian and Hamiltonian Descriptions of Evolution Equations1.2.9 The Lagrangian Descriptions of the Harmonic Oscillator; 1.2.10 Constructing Nonlinear Systems Out of Linear Ones; 1.2.11 The Reparametrized Harmonic Oscillator; 1.2.12 Reduction of Linear Systems; 1.3 Linear Systems with Infinite Degrees of Freedom; 1.3.1 The Klein-Gordon Equation and the Wave Equation; 1.3.2 The Maxwell Equations; 1.3.3 The Schr©œdinger Equation; 1.3.4 Symmetries and Infinite-Dimensional Systems; 1.3.5 Constants of Motion; References
2 The Language of Geometry and Dynamical Systems: The Linearity Paradigm2.1 Introduction; 2.2 Linear Dynamical Systems: The Algebraic Viewpoint; 2.2.1 Linear Systems and Linear Spaces; 2.2.2 Integrating Linear Systems: Linear Flows; 2.2.3 Linear Systems and Complex Vector Spaces; 2.2.4 Integrating Time-Dependent Linear Systems: Dyson's Formula; 2.2.5 From a Vector Space to Its Dual: Induced Evolution Equations; 2.3 From Linear Dynamical Systems to Vector Fields; 2.3.1 Flows in the Algebra of Smooth Functions; 2.3.2 Transformations and Flows
2.3.3 The Dual Point of View of Dynamical Evolution2.3.4 Differentials and Vector Fields: Locality; 2.3.5 Vector Fields and Derivations on the Algebra of Smooth Functions; 2.3.6 The `Heisenberg' Representation of Evolution; 2.3.7 The Integration Problem for Vector Fields; 2.4 Exterior Differential Calculus on Linear Spaces; 2.4.1 Differential Forms; 2.4.2 Exterior Differential Calculus: Cartan Calculus; 2.4.3 The `Easy' Tensorialization Principle; 2.4.4 Closed and Exact Forms; 2.5 The General `Integration' Problem for Vector Fields
2.5.1 The Integration Problem for Vector Fields: Frobenius Theorem2.5.2 Foliations and Distributions; 2.6 The Integration Problem for Lie Algebras; 2.6.1 Introduction to the Theory of Lie Groups: Matrix Lie Groups; 2.6.2 The Integration Problem for Lie Algebras*; References; 3 The Geometrization of Dynamical Systems; 3.1 Introduction; 3.2 Differentiable Spaces; 3.2.1 Ideals and Subsets; 3.2.2 Algebras of Functions and Differentiable Algebras; 3.2.3 Generating Sets; 3.2.4 Infinitesimal Symmetries and Constants of Motion; 3.2.5 Actions of Lie Groups and Cohomology
1.2.8 Lagrangian and Hamiltonian Descriptions of Evolution Equations1.2.9 The Lagrangian Descriptions of the Harmonic Oscillator; 1.2.10 Constructing Nonlinear Systems Out of Linear Ones; 1.2.11 The Reparametrized Harmonic Oscillator; 1.2.12 Reduction of Linear Systems; 1.3 Linear Systems with Infinite Degrees of Freedom; 1.3.1 The Klein-Gordon Equation and the Wave Equation; 1.3.2 The Maxwell Equations; 1.3.3 The Schr©œdinger Equation; 1.3.4 Symmetries and Infinite-Dimensional Systems; 1.3.5 Constants of Motion; References
2 The Language of Geometry and Dynamical Systems: The Linearity Paradigm2.1 Introduction; 2.2 Linear Dynamical Systems: The Algebraic Viewpoint; 2.2.1 Linear Systems and Linear Spaces; 2.2.2 Integrating Linear Systems: Linear Flows; 2.2.3 Linear Systems and Complex Vector Spaces; 2.2.4 Integrating Time-Dependent Linear Systems: Dyson's Formula; 2.2.5 From a Vector Space to Its Dual: Induced Evolution Equations; 2.3 From Linear Dynamical Systems to Vector Fields; 2.3.1 Flows in the Algebra of Smooth Functions; 2.3.2 Transformations and Flows
2.3.3 The Dual Point of View of Dynamical Evolution2.3.4 Differentials and Vector Fields: Locality; 2.3.5 Vector Fields and Derivations on the Algebra of Smooth Functions; 2.3.6 The `Heisenberg' Representation of Evolution; 2.3.7 The Integration Problem for Vector Fields; 2.4 Exterior Differential Calculus on Linear Spaces; 2.4.1 Differential Forms; 2.4.2 Exterior Differential Calculus: Cartan Calculus; 2.4.3 The `Easy' Tensorialization Principle; 2.4.4 Closed and Exact Forms; 2.5 The General `Integration' Problem for Vector Fields
2.5.1 The Integration Problem for Vector Fields: Frobenius Theorem2.5.2 Foliations and Distributions; 2.6 The Integration Problem for Lie Algebras; 2.6.1 Introduction to the Theory of Lie Groups: Matrix Lie Groups; 2.6.2 The Integration Problem for Lie Algebras*; References; 3 The Geometrization of Dynamical Systems; 3.1 Introduction; 3.2 Differentiable Spaces; 3.2.1 Ideals and Subsets; 3.2.2 Algebras of Functions and Differentiable Algebras; 3.2.3 Generating Sets; 3.2.4 Infinitesimal Symmetries and Constants of Motion; 3.2.5 Actions of Lie Groups and Cohomology