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Mathematical methods of classical mechanics

Author: V I Arnolʹd
Publisher: New York : Springer-Verlag, ©1978.
Series: Graduate texts in mathematics, 60.
Edition/Format:   eBook : Document : EnglishView all editions and formats
Summary:
Many different mathematical methods and concepts are used in classical mechanics: differential equations and phase ftows, smooth mappings and manifolds, Lie groups and Lie algebras, symplectic geometry and ergodic theory. Many modern mathematical theories arose from problems in mechanics and only later acquired that axiomatic-abstract form which makes them so hard to study. In this book we construct the mathematical  Read more...
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Genre/Form: Electronic books
Additional Physical Format: Print version:
Arnolʹd, V.I. (Vladimir Igorevich), 1937-2010.
Mathematical methods of classical mechanics.
New York : Springer-Verlag, ©1978
(DLC) 78015927
(OCoLC)4037141
Material Type: Document, Internet resource
Document Type: Internet Resource, Computer File
All Authors / Contributors: V I Arnolʹd
ISBN: 9781475716931 1475716931 9781475716955 1475716958
OCLC Number: 681897672
Reproduction Notes: Electronic reproduction. [Place of publication not identified] : HathiTrust Digital Library, 2010. MiAaHDL
Description: 1 online resource (x, 462 pages) : illustrations
Details: Master and use copy. Digital master created according to Benchmark for Faithful Digital Reproductions of Monographs and Serials, Version 1. Digital Library Federation, December 2002.
Contents: I Newtonian Mechanics --
1 Experimental facts --
2 Investigation of the equations of motion --
II Lagrangian Mechanics --
3 Variational principles --
4 Lagrangian mechanics on manifolds --
5 Oscillations --
6 Rigid Bodies --
III Hamiltonian Mechanics --
7 Differential forms --
8 Symplectic manifolds --
9 Canonical formalism --
10 Introduction to perturbation theory --
Appendix 1 Riemannian curvature --
Appendix 2 Geodesies of left-invariant metrics on Lie groups and the hydrodynamics of an ideal fluid --
Appendix 3 Symplectic structure on algebraic manifolds --
Appendix 4 Contact structures --
Appendix 5 Dynamical systems with symmetries --
Appendix 6 Normal forms of quadratic hamiltonians --
Appendix 7 Normal forms of hamiltonian systems near stationary points and closed trajectories --
Appendix 8 Perturbation theory of conditionally periodic motions and Kolmogorov's theorem --
Appendix 9 Poincaré's geometric theorem, its generalizations and applications --
Appendix 10 Multiplicities of characteristic frequencies, and ellipsoids depending on parameters --
Appendix 11 Short wave asymptotics --
Appendix 12 Lagrangian singularities --
Appendix 13 The Korteweg-de Vries equation.
Series Title: Graduate texts in mathematics, 60.
Other Titles: Matematicheskie metody klassicheskoĭ mekhaniki.
Responsibility: V.I. Arnold ; translated by K. Vogtmann and A. Weinstein.

Abstract:

Many different mathematical methods and concepts are used in classical mechanics: differential equations and phase ftows, smooth mappings and manifolds, Lie groups and Lie algebras, symplectic geometry and ergodic theory. Many modern mathematical theories arose from problems in mechanics and only later acquired that axiomatic-abstract form which makes them so hard to study. In this book we construct the mathematical apparatus of classical mechanics from the very beginning; thus, the reader is not assumed to have any previous knowledge beyond standard courses in analysis (differential and integral calculus, differential equations), geometry (vector spaces, vectors) and linear algebra (linear operators, quadratic forms). With the help of this apparatus, we examine all the basic problems in dynamics, including the theory of oscillations, the theory of rigid body motion, and the hamiltonian formalism. The author has tried to show the geometric, qualitative aspect of phenomena. In this respect the book is closer to courses in theoretical mechanics for theoretical physicists than to traditional courses in theoretical mechanics as taught by mathematicians.

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