Fundamental Principle of Classical Mechanics - A Geometrical Perspective

Description

Contents Preface 1 Vectors, Tensors, and Linear Transformations 2 Exterior Algebra: Determinants, Oriented Frames and Oriented Volumes 3 The Hodge-Star Operator and the Vector Cross Product 4 Kinematics and Moving Frames: From the Angular Velocity to Gauge Fields 5 Differentiable Manifolds: The Tangent and Cotangent Bundles 6 Exterior Calculus: Differential Forms 7 Vector Calculus by Differential Forms 8 The Stokes Theorem 9 Cartan’s Method of Moving Frames: Curvilinear Coordinates in R3 10 Mechanical Constraints: The Frobenius Theorem 11 Flows and Lie Derivatives 12 Newton’s Laws: Inertial and Non-inertial Frames 13 Simple Applications of Newton’s Laws 14 Potential Theory: Newtonian Gravitation 15 Centrifugal and Coriolis Forces 16 Harmonic Oscillators: Fourier Transforms and Green’s Functions 17 Classical Model of the Atom: Power Spectra 18 Dynamical Systems and their Stabilities 19 Many-Particle Systems and the Conservation Principles 20 Rigid-Body Dynamics: The Euler-Poisson Equations of Motion 21 Topology and Systems with Holonomic Constraints: Homology and de Rham Cohomology 22 Connections on Vector Bundles: Affine Connections on Tangent Bundles 23 The Parallel Translation of Vectors: The Foucault Pendulum 24 Geometric Phases, Gauge Fields, and the Mechanics of Deformable Bodies: The “Falling Cat” Problem 25 Force and Curvature 26 The Gauss-Bonnet-Chern Theorem and Holonomy 27 The Curvature Tensor in Riemannian Geometry 28 Frame Bundles and Principal Bundles, Connections on Principal Bundles 29 Calculus of Variations, the Euler-Lagrange Equations, the First Variation of Arclength and Geodesics 30 The Second Variation of Arclength, Index Forms, and Jacobi Fields 31 The Lagrangian Formulation of Classical Mechanics: Hamilton’s Principle of Least Action, Lagrange Multipliers in Constrained Motion 32 Small Oscillations and Normal Modes 33 The Hamiltonian Formulation of Classical Mechanics: Hamilton’s Equations of Motion 34 Symmetry and Conservation 35 Symmetric Tops 36 Canonical Transformations and the Symplectic Group 37 Generating Functions and the Hamilton-Jacobi Equation 38 Integrability, Invariant Tori, Action-Angle Variables 39 Symplectic Geometry in Hamiltonian Dynamics, Hamiltonian Flows, and Poincare-Cartan Integral Invariants 40 Darboux’s Theorem in Symplectic Geometry 41 The Kolmogorov-Arnold-Moser (KAM) Theorem 42 The Homoclinic Tangle and Instability, Shifts as Subsystems 43 The Restricted Three-Body Problem References Index