Mathematical Aspects of Classical and Celestial MechanicsIn this book we describe the basic principles, problems, and methods of cl- sical mechanics. Our main attention is devoted to the mathematical side of the subject. Although the physical background of the models considered here and the applied aspects of the phenomena studied in this book are explored to a considerably lesser extent, we have tried to set forth ?rst and foremost the “working” apparatus of classical mechanics. This apparatus is contained mainly in Chapters 1, 3, 5, 6, and 8. Chapter 1 is devoted to the basic mathematical models of classical - chanics that are usually used for describing the motion of real mechanical systems. Special attention is given to the study of motion with constraints and to the problems of realization of constraints in dynamics. In Chapter 3 we discuss symmetry groups of mechanical systems and the corresponding conservation laws. We also expound various aspects of ord- reduction theory for systems with symmetries, which is often used in appli- tions. Chapter 4 is devoted to variational principles and methods of classical mechanics. They allow one, in particular, to obtain non-trivial results on the existence of periodic trajectories. Special attention is given to the case where the region of possible motion has a non-empty boundary. Applications of the variational methods to the theory of stability of motion are indicated. |
Contents
The nBody Problem | 61 |
Symmetry Groups and Order Reduction | 103 |
Variational Principles and Methods 135 | 134 |
Integrable Systems and Integration Methods | 171 |
6 | 207 |
NonIntegrable Systems | 351 |
Theory of Small Oscillations 401 | 400 |
Tensor Invariants of Equations of Dynamics | 431 |
| 471 | |
| 507 | |
Other editions - View all
Mathematical Aspects of Classical and Celestial Mechanics Vladimir I. Arnold,Valery V. Kozlov,Anatoly I. Neishtadt No preview available - 2009 |
Mathematical Aspects of Classical and Celestial Mechanics Vladimir I. Arnold,Valery V. Kozlov,Anatoly I. Neishtadt No preview available - 2010 |
Mathematical Aspects of Classical and Celestial Mechanics Vladimir I. Arnold,Valery V. Kozlov,Anatoly I. Neishtadt No preview available - 2006 |
Common terms and phrases
action adiabatic analytic applied approximation assume asymptotic averaged body called canonical centre change of variables closed complete condition connected Consequently consider const constant constraints contained coordinates corresponding curve defined definition degrees of freedom depends described differential domain energy equal equations example existence field fixed force formula frequencies function given Hamiltonian system holds independent initial integral introduce invariant Lagrangian linear manifold mass measure mechanics method momentum motion natural neighbourhood non-degenerate normal obtain orbits original oscillations parameter passing periodic perturbed phase plane Poincaré potential principle problem proof Proposition proved quantity reduced relations resonance respect rotation satisfy separatrix slow smooth solution stable Suppose surface symmetry symplectic takes Theorem theory tion torus trajectories transformation unperturbed variables variation vector zero
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