Mathematical Methods of Classical MechanicsSpringer Science & Business Media, 2013 M11 11 - 464 páginas 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. |
Contenido
2 | |
14 | |
15 | |
Investigation of motion in a central field | 33 |
Motions of a system of n points | 44 |
The method of similarity | 50 |
Liouvilles theorem | 68 |
Holonomic constraints | 85 |
Generating functions | 266 |
Actionangle variables | 279 |
Averaging | 285 |
Averaging of perturbations | 291 |
Appendix 1 | 301 |
Appendix 2 | 310 |
Geodesics of leftinvariant metrics on Lie groups and | 318 |
Symplectic structure on algebraic manifolds | 346 |
Linearization | 98 |
91 | 111 |
Inertial forces and the Coriolis force | 129 |
Eulers equations Poinsots description of the motion | 142 |
Part III | 160 |
Appendix 9 | 178 |
Integration of differential forms | 181 |
Symplectic geometry | 219 |
Parametric resonance in systems with many degrees of freedom | 225 |
Chapter 9 | 233 |
Applications of the integral invariant of PoincaréCartan | 240 |
Huygens principle | 248 |
The HamiltonJacobi method for integrating Hamiltons canonical | 258 |
Appendix 5 | 371 |
Appendix 6 | 381 |
Appendix 8 | 393 |
Perturbation theory of conditionally periodic motions | 399 |
Chapter 8 | 401 |
Poincarés geometric theorem its generalizations | 416 |
Appendix 11 | 438 |
Calculus of variations | 452 |
Appendix 13 | 453 |
458 | |
459 | |
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action-angle variables angular momentum angular velocity axis called canonical transformation characteristic frequencies closed configuration space consider contact manifold coordinate system Corollary corresponding cotangent bundle curvature defined Definition degrees of freedom denote diffeomorphism differential equations dimension direction eigenvalues ellipse ellipsoid equal to zero equilibrium position euclidean space example Figure formula function H geodesic given H dt hamiltonian function I₁ inertia inertia ellipsoid initial conditions integral invariant tori k-form kinetic energy lagrangian manifold Legendre Lemma Lie algebra linear mapping metric n-dimensional neighborhood nondegenerate normal form obtain orbit P₁ parameter pendulum perturbation phase curves phase flow phase space plane Poisson bracket potential energy PROBLEM PROOF q₁ quadratic form resonance riemannian manifold rigid body rotation Show singularities solution stationary submanifold surface symmetric symplectic manifold symplectic structure tangent space theorem three-dimensional torus trajectories two-dimensional vector field vector space