Mathematical methods of classical mechanics
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
1 online resource (x, 462 pages) : illustrations
9781475716931, 9781475716955, 1475716931, 1475716958
681897672
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
Electronic reproduction, [Place of publication not identified], HathiTrust Digital Library, 2010
Text in English, translated from the original Russian