经典力学和天体力学中的数学论题 第2版
2000-6
世界图书出版公司
V.I.Arnold
291
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This work describes the fundamental principles, problems, and methods of classical mechanics focussing on its mathematical aspects. The authors have striven to give an exposition stressing the working apparatus of classical mechanics, rather than its physical foundations or applications. This apparatus is basically contained in Chapters 1, 3, 4 and 5. Chapter 1 is devoted to the fundamental mathematical models which are usually employed to describe the motion of real mechanical systems. Special consideration is given to the study of motion under constraints, and also to problems concerned with the realization of constraints in dynamics. This work describes the fundamental principles, problems, and methods of classical mechanics focussing on its mathematical aspects. The authors have striven to give an exposition stressing the working apparatus of classical mechanics, rather than its physical foundations or applications. This apparatus is basically contained in Chapters 1, 3, 4 and 5. Chapter 1 is devoted to the fundamental mathematical models which are usually employed to describe the motion of real mechanical systems. Special consideration is given to the study of motion under constraints, and also to problems concerned with the realization of constraints in dynamics.
Chapter 1. Basic Principles of Classical Mechanics 1. Newtonian Mechanics 1.1. Space. Time, Motion 1.2. The Newton-Laplace Principle of Determinacy 1.3. The Principle of Relativity 1.4. Basic Dynamical Quantities. Conservation Laws 2. Lagrangian Mechanics 2.1. Preliminary Remarks 2.2. Variations and Extremals 2.3. Lagrange''s Equations 2.4. Poincare''s Equations 2.5. Constrained Motion 3. Hamiltonian Mechanics 3.1. Symplectic Structures and Hamilton''s Equations 3.2. Generating Functions 3.3. Symplectic Structure of the Cotangent Bundle 3.4. The Problem of n Point Vortices 3.5. The Action Functional in Phase Space 3.6. Integral Invariants 3.7. Applications to the Dynamics of Ideal Fluids 3.8. Principle of Stationary Isoenergetic Action 4. Vakonomic Mechanics 4.1. Lagrange''s Problem 4.2. Vakonomic Mechanics ……Chapter 2. The n-Body ProblemChapter 3. Symmetry Groups and Reduction Lowering the OrderChapter 4. Integrable Systems and Integration MethodsChapter 5. Perturbation Theory for Integrable SystemsChapter 6. Nonintegrable SystemsChapter 7. Theory of Small OscillationsComments on the BibliographyRecommended ReadingBibliographyIndex
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