Table of Contents

Vectors, Tensors, and Linear Transformations; The Hodge - Star Operator and the Vector Cross Product; Differentiable Manifolds: the Tangent and Cotangent Bundles; Vector Calculus by Differential Forms; Cartan's Method of Moving Frames: Curvilinear Coordinates in R3; Flows and Lie Derivatives; Simple Applications of Newton's Laws; Centrifugal and Coriolis Forces; Classical Model of the Atom: Power Spectra; Many-Particle Systems and the Conservation Principles; Topology and Systems with Holonomic Constraints: Homology and de Rham Cohomology; The Parallel Transport of Vectors: The Foucault Pendulum; Force and Curvature; The Curvature Tensor in Riemannian Geometry; Calculus of Variations, the Euler - Lagrange Equations, the First Variation of Arc Length and Geodesics; The Second Variation of Arc Length, Index Forms, and Jacobi Fields; The Lagrangian Formulation of Classical Mechanics: Hamilton's Principle of Least Action, Lagrange Multipliers in Constrained Motion; The Hamiltonian Formulation of Classical Mechanics: Hamilton's Equations of Motion; Symmetric Tops; Integrability, Invariant Tori, Action-Angle Variables; Hamilton - Jacobi Theory, Integral Invariants; The Kolmogorov - Arnold - Moser (KAM) Theory: Stability of Invariant Tori; The Three-Body Problem; and other chapters.