Fundamental Principle of Classical Mechanics - A Geometrical Perspective

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