Fundamental Principles of Classical Mechanics

This book is written with the belief that classical mechanics, as a theoretical discipline, possesses an inherent beauty, depth, and richness that far transcends its immediate applications in mechanical systems. These properties are manifested, by and large, through the coherence and elegance of the mathematical structure underlying the discipline, and are eminently worthy of being communicated to physics students at the earliest stage possible. This volume is therefore addressed mainly to advanced undergraduate and beginning graduate physics students who are interested in the application of modern mathematical methods in classical mechanics, in particular, those derived from the fields of topology and differential geometry, and also to the occasional mathematics student who is interested in important physics applications of these areas of mathematics. Its main purpose is to offer an introductory and broad glimpse of the majestic edifice of the mathematical theory of classical dynamics, not only in the time-honored analytical tradition of Newton, Laplace, Lagrange, Hamilton, Jacobi, and Whittaker, but also the more topological/geometrical one established by Poincare, and enriched by Birkhoff, Lyapunov, Smale, Siegel, Kolmogorov, Arnold, and Moser (as well as many others).Contents: Vectors, Tensors, and Linear Transformations Exterior Algebra: Determinants, Oriented Frames and Oriented Volumes The Hodge–Star Operator and the Vector Cross Product Kinematics and Moving Frames: From the Angular Velocity to Gauge Fields Differentiable Manifolds: The Tangent and Cotangent Bundles Exterior Calculus: Differential Forms Vector Calculus by Differential Forms The Stokes Theorem Cartan's Method of Moving Frames: Curvilinear Coordinates in ℝ3 Mechanical Constraints: The Frobenius Theorem Flows and Lie Derivatives Newton's Laws: Inertial and Non-inertial Frames Simple Applications of Newton's Laws Potential Theory: Newtonian Gravitation Centrifugal and Coriolis Forces Harmonic Oscillators: Fourier Transforms and Green's Functions Classical Model of the Atom: Power Spectra Dynamical Systems and Their Stabilities Many-Particle Systems and the Conservation Principles Rigid-Body Dynamics: The Euler-Poisson Equations of Motion Topology and Systems with Holonomic Constraints: Homology and de Rham Cohomology Connections on Vector Bundles: Affine Connections on Tangent Bundles The Parallel Translation of Vectors: The Foucault Pendulum Geometric Phases, Gauge Fields, and the Mechanics of Deformable Bodies: The “Falling Cat” Problem Force and Curvature The Gauss-Bonnet-Chern Theorem and Holonomy The Curvature Tensor in Riemannian Geometry Frame Bundles and Principal Bundles, Connections on Principal Bundles Calculus of Variations, the Euler-Lagrange Equations, the First Variation of Arclength and Geodesics The Second Variation of Arclength, Index Forms, and Jacobi Fields The Lagrangian Formulation of Classical Mechanics: Hamilton's Principle of Least Action, Lagrange Multipliers in Constrained Motion Small Oscillations and Normal Modes The Hamiltonian Formulation of Classical Mechanics: Hamilton's Equations of Motion Symmetry and Conservation Symmetric Tops Canonical Transformations and the Symplectic Group Generating Functions and the Hamilton-Jacobi Equation Integrability, Invariant Tori, Action-Angle Variables Symplectic Geometry in Hamiltonian Dynamics, Hamiltonian Flows, and Poincaré-Cartan Integral Invariants Darboux's Theorem in Symplectic Geometry The Kolmogorov-Arnold-Moser (KAM) Theorem The Homoclinic Tangle and Instability, Shifts as Subsystems The Restricted Three-Body Problem Readership: Advanced undergraduate and beginning graduate students in classical mechanics, mathematical physics and differential geometry.