Differential Geometry, Differential Equations, and Mathematical Physics

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Preface Acknowledgments Contents Contributors 1 Poisson and Symplectic Structures, Hamiltonian Action, Momentum and Reduction 1.1 Introduction 1.2 Poisson and Symplectic Structures 1.2.1 Poisson Manifolds 1.2.2 Poisson Bi-vector 1.2.2.1 Hamilton Map and Jacobi Identity 1.2.3 Symplectic Structures on Manifolds 1.2.3.1 Darboux Theorem and Hamiltonian Vector Fields 1.2.3.2 Example 1.2.4 Co-tangent Bundle: Liouville Form 1.2.5 Non-Symplectic Poisson Structures 1.2.6 Poisson Brackets on Dual of a Lie Algebra 1.2.6.1 Definition via Gradient Operator 1.2.6.2 Definition via Canonical Structure on T(G) 1.3 Group Actions and Orbits 1.3.1 Stabilizers and Orbits 1.3.2 Infinitesimal Action 1.3.3 Lie Group and Lie Algebra Representations 1.3.3.1 Adjoint Representations 1.3.3.2 Co-Adjoint Representations 1.4 Moment Map, Poisson and Hamiltonian Actions 1.4.1 Introductory Motivation 1.4.2 Momentum Map 1.4.3 Moment and Hamiltonian Actions 1.4.3.1 Examples 1.5 Reduction of the Phase Space 1.5.1 The Main Results 1.5.2 Example 1.6 Poisson–Lie Groups 1.6.1 Modified Classical Yang–Baxter Equation 1.6.2 Manin Triples 1.6.3 Poisson–Lie Duality 1.6.4 Example of Non-Hamiltonian Action References 2 Notes on Tractor Calculi 2.1 Tracy Thomas' Conformal Tractors 2.1.1 Riemannian Sphere 2.1.2 Conformal Riemannian Sphere 2.1.3 Towards Tractors 2.1.4 Tracy Thomas' Tractors T 2.2 Conformal to Einstein and the Tractor Connection 2.2.1 The Einstein Scales 2.2.2 Conformal Invariance 2.3 Parabolic Geometries 2.3.1 |1|-graded Parabolic Geometries 2.3.2 Natural Bundles and Weyl Connections 2.3.3 Higher Order Derivatives 2.3.4 A Few Examples 2.4 Elements of Tractor Calculus 2.4.1 Natural Bundles and Tractors 2.4.2 Adjoint Tractors 2.4.3 Fundamental Derivative 2.4.4 Back to Parabolic Geometries 2.4.5 Towards Effective Calculus for Conformal Geometry 2.5 The (Co)homology and Normalization 2.5.1 Deformations of Cartan Connections 2.5.2 Homology and Cohomology 2.5.3 Normalization of Parabolic Geometries 2.6 The BGG Machinery 2.6.1 The Twisted de-Rham Complexes 2.6.2 BGG Machinery 2.6.3 The First BGG Operators References 3 Symmetries and Integrals 3.1 Preface 3.2 Distributions 3.3 Distributions and Differential Equations 3.3.1 Cartan Distributions (ODEs) 3.3.2 Cartan Distributions (PDEs) 3.4 Symmetry 3.4.1 Symmetries of the Cartan Distributions 3.4.2 Symmetries of Completely Integrable Distributions 3.5 The Lie–Bianchi Theorem 3.5.1 Commutative Lie Algebra Symmetries 3.5.2 Symmetry Reduction 3.5.3 Quadratures and Model Equations 3.5.4 The Lie Superposition Principle 3.6 Ordinary Differential Equations 3.7 ODE Symmetries 3.7.1 Integration of ODEs with Commutative Symmetry Algebras 3.8 Schrödinger Type Equations 3.8.1 Actions of Diffeomorphisms on Schrödinger Type Equations 3.8.2 Actions of the Diffeomorphism Group on Tensors 3.8.3 Integration of Schrödinger Type Equations with Integrable Potentials 3.8.3.1 Integration by Symmetries 3.8.3.2 Lame Equation 3.8.3.3 Eigenvalue Problem References 4 Finite Dimensional Dynamics of Evolutionary Equationswith Maple 4.1 Introduction 4.2 Symmetries of ODEs 4.3 Flows on ODE's Solution Spaces 4.4 The Fisher–Kolmogorov–Petrovsky–Piskunov Equation 4.4.1 Second-Order Dynamics 4.4.2 Integration of the Dynamics 4.4.3 Construction Solutions of the FKPP Equation by Dynamics 4.5 The Reaction–Diffusion Equation with a Convection Term 4.5.1 First-Order Dynamics 4.5.2 Second-Order Dynamics 4.6 The Black–Scholes Equation 4.6.1 First-Order Dynamics 4.6.2 Construction Solutions of the Black–Scholes Equation by Dynamics References 5 Critical Phenomena in Darcy and Euler Flows of Real Gases 5.1 Introduction 5.2 Measurement and Thermodynamics 5.2.1 Measurement of Random Vectors 5.2.2 Information Gain 5.2.3 The Principle of Minimal Information Gain 5.2.4 Variance of Random Vectors 5.2.5 Thermodynamics 5.3 Thermodynamics of Gases 5.3.1 Specific Variables 5.3.2 Legendrian and Lagrangian Manifolds for Gases 5.3.3 Singularities of Lagrangian Manifolds and Phase Transitions 5.4 Examples of Gases 5.4.1 Ideal Gases 5.4.2 Van der Waals Gases 5.4.3 Peng–Robinson Gases 5.4.4 Redlich–Kwong Gases 5.5 Basic Equations 5.5.1 Darcy Flows 5.5.2 Euler Flows 5.6 Solutions 5.6.1 Darcy Flows 5.6.1.1 Ideal Gases 5.6.1.2 van der Waals Gases 5.6.1.3 Peng–Robinson Gases 5.6.1.4 Redlich–Kwong Gases 5.6.2 Euler Flows 5.6.2.1 Ideal Gases 5.6.2.2 van der Waals Gases 5.7 Conclusions References 6 Differential Invariants for Flows of Fluids and Gases 6.1 Introduction 6.2 Thermodynamics 6.3 Compressible Inviscid Fluids or Gases 6.3.1 2D-Flows 6.3.1.1 Symmetry Lie Algebra 6.3.1.2 Symmetry Classification of States 6.3.1.3 Differential Invariants 6.3.2 Flows on a Sphere 6.3.2.1 Symmetry Lie Algebra 6.3.2.2 Symmetry Classification of States 6.3.2.3 Differential Invariants 6.3.3 Flows on a Spherical Layer 6.3.3.1 Symmetry Lie Algebra 6.3.3.2 Symmetry Classification of States 6.3.3.3 Differential Invariants 6.4 Compressible Viscid Fluids or Gases 6.4.1 2D-Flows 6.4.1.1 Symmetry Lie Algebra 6.4.1.2 Symmetry Classification of States 6.4.1.3 Differential Invariants 6.4.2 3D-Flows 6.4.2.1 Symmetry Lie Algebra 6.4.2.2 Symmetry Classification of States 6.4.2.3 Differential Invariants 6.4.3 Flows on a Sphere 6.4.3.1 Symmetry Lie Algebra 6.4.3.2 Symmetry Classification of States 6.4.3.3 Differential Invariants 6.4.4 Flows on a Spherical Layer 6.4.4.1 Symmetry Lie Algebra 6.4.4.2 Symmetry Classification of States 6.4.4.3 Differential Invariants 6.4.5 The Field of Navier–Stokes Invariants References