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 | |
| 15 | |
Motions of a system of n points | 44 |
Legendre transformations | 61 |
Chapter 4 | 75 |
E Noethers theorem | 88 |
Small oscillations | 103 |
Chapter 6 | 123 |
The HamiltonJacobi method for integrating Hamiltons canonical | 258 |
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 |
Eulers equations Poinsots description of the motion | 142 |
Part III | 160 |
Appendix 9 | 178 |
Integration of differential forms | 181 |
Chapter 8 | 201 |
The Lie algebra of hamiltonian functions | 214 |
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 |
Appendix 4 | 333 |
Contact structures | 349 |
Appendix 5 | 371 |
Appendix 6 | 381 |
Appendix 8 | 393 |
Perturbation theory of conditionally periodic motions | 399 |
Poincarés geometric theorem its generalizations | 416 |
Appendix 11 | 438 |
Appendix 13 | 453 |
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action-angle variables angular velocity axis called canonical transformation configuration space consider contact elements contact manifold contact structure coordinate system Corollary corresponding cotangent bundle curvature defined Definition denote diffeomorphism differential equations differential form dimension direction e₁ eigenvalues ellipsoid equal to zero equilibrium position euclidean space example F₁ Figure formula function H geodesic given group G H dt hamiltonian function hyperplanes initial conditions k-form Legendre Lemma Lie algebra Lie group linear M₁ mapping metric motion n-dimensional neighborhood nondegenerate normal form obtain one-parameter orbit oriented oscillations P₁ parameter perturbation phase curves phase flow phase space plane Poisson bracket potential energy PROBLEM PROOF q₁ resonance riemannian manifold rigid body rotation singularities solution stable stationary submanifold surface symplectic manifold symplectic structure tangent space theorem three-dimensional torus two-dimensional variables vector field vector space x₁ ω²