Fundamental principles of classical mechanics : a geometrical perspective
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書誌事項
Fundamental principles of classical mechanics : a geometrical perspective
World Scientific, c2014
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注記
Includes bibliographical references and index
内容説明・目次
内容説明
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).
目次
- 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.
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