Fundamental Principles of Classical Mechanics: A Geometrical Perspective

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 TransformationsExterior Algebra: Determinants, Oriented Frames and Oriented VolumesThe Hodge–Star Operator and the Vector Cross ProductKinematics and Moving Frames: From the Angular Velocity to Gauge FieldsDifferentiable Manifolds: The Tangent and Cotangent BundlesExterior Calculus: Differential FormsVector Calculus by Differential FormsThe Stokes TheoremCartan's Method of Moving Frames: Curvilinear Coordinates in ℝ3Mechanical Constraints: The Frobenius TheoremFlows and Lie DerivativesNewton's Laws: Inertial and Non-inertial FramesSimple Applications of Newton's LawsPotential Theory: Newtonian GravitationCentrifugal and Coriolis ForcesHarmonic Oscillators: Fourier Transforms and Green's FunctionsClassical Model of the Atom: Power SpectraDynamical Systems and Their StabilitiesMany-Particle Systems and the Conservation PrinciplesRigid-Body Dynamics: The Euler-Poisson Equations of MotionTopology and Systems with Holonomic Constraints: Homology and de Rham CohomologyConnections on Vector Bundles: Affine Connections on Tangent BundlesThe Parallel Translation of Vectors: The Foucault PendulumGeometric Phases, Gauge Fields, and the Mechanics of Deformable Bodies: The “Falling Cat” ProblemForce and CurvatureThe Gauss-Bonnet-Chern Theorem and HolonomyThe Curvature Tensor in Riemannian GeometryFrame Bundles and Principal Bundles, Connections on Principal BundlesCalculus of Variations, the Euler-Lagrange Equations, the First Variation of Arclength and GeodesicsThe Second Variation of Arclength, Index Forms, and Jacobi FieldsThe Lagrangian Formulation of Classical Mechanics: Hamilton's Principle of Least Action, Lagrange Multipliers in Constrained MotionSmall Oscillations and Normal ModesThe Hamiltonian Formulation of Classical Mechanics: Hamilton's Equations of MotionSymmetry and ConservationSymmetric TopsCanonical Transformations and the Symplectic GroupGenerating Functions and the Hamilton-Jacobi EquationIntegrability, Invariant Tori, Action-Angle VariablesSymplectic Geometry in Hamiltonian Dynamics, Hamiltonian Flows, and Poincaré-Cartan Integral InvariantsDarboux's Theorem in Symplectic GeometryThe Kolmogorov-Arnold-Moser (KAM) TheoremThe Homoclinic Tan