Mathematical Methods of Classical MechanicsSpringer New York, 1978 - 462 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
Part | 1 |
Chapter | 15 |
Motions of a system of n points | 44 |
Derechos de autor | |
Otras 34 secciones no mostradas
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Términos y frases comunes
angular momentum angular velocity axis called canonical transformation configuration space consider contact elements contact manifold contact structure coordinate system Corollary corresponding cotangent bundle curvature defined Definition degrees of freedom denote diffeomorphism differential equations differential form dimension direction e₁ eigenvalues ellipse ellipsoid equal to zero equilibrium position euclidean space example F₁ Figure formula frequencies function H geodesic given group G H dt hamiltonian function hyperplanes initial conditions k-form kinetic energy Legendre Lemma Lie algebra linear mapping metric 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 Show solution stationary submanifold surface symplectic manifold symplectic structure tangent space theorem three-dimensional torus two-dimensional variables vector field vector space velocity vector