Нашли опечатку? Выделите ее мышкой и нажмите Ctrl+Enter
Название: Ordinary Differential Equations
Автор: Arnold V.I.
Аннотация:
Although there is no lack of other books on this subject, even with the same title, the appearance of this new one is fully justified on at least two grounds: its approach makes full use of modern mathematical concepts and terminology of considerable sophistication and abstraction, going well beyond the traditional presentation of the subject; and, at the same time, the resulting enhancement of mathematical abstractness is counterbalanced by a constant appeal to geometrical and physical considerations, presented in the main text and in numerous problems and exercises.
In terms of mathematical approach, the text is dominated by two central ideas: the theorem on rectifiability of a vector field (which is equivalent to the usual theorems on existence, uniqueness, and differentiability of solutions) and the theory of one-parameter groups of linear transformations (equivalent to the theory of linear autonomous systems). The book also develops whole congeries of fundamental concepts—like phase space and phase flows, smooth manifolds and tangent bundles, vector fields and one-parameter groups of diffeomorphisms—that remain in the shadows in the traditional coordinate-based approach. All of these concepts are presented in some detail, but without assuming any background on the part of the reader beyond the scope of the standard elementary courses on analysis and linear algebra.
In terms of concrete applications, the book introduces the pendulum equation at the very beginning, and the efficacy of various concepts and methods introduced throughout is subsequently tested by applying them to this example. Thus, the section on first integrals leads to the law of conservation of energy; the theorem on differentiation with respect to a parameter finds application in the "method of small parameters"; and the theory of linear equations with periodic coefficients merges naturally with the study of parametric resonance. This geometrical and physical specificity is made still more vivid through the inclusion of 259 line drawings and 260 exercises in which other examples are taken up. — This text refers to an out of print or unavailable edition of this title.
Node25 Nonautonomous linear equation188—189 Nonautonomous linear equation of order n193 Nonautonomous linear equation of order n, fundamental system of solutions of193 Nonautonomous linear equation of order n, space of solutions of193 Nonautonomous linear equation, fundamental system of solutions of191 Nonautonomous linear equation, space of solutions of191 Nonautonomous system31 Norm of a vector215 Norm of an operator98216 Normal form67 Normal modes175 Normed linear space99 Normed linear space, norm of element of99 Null section245 Number of revolutionssee "Winding number" One-parameter group of diffeomorphisms620 One-parameter group of linear transformations2096106 One-parameter group of transformations4 Oriented manifold235 Oscillations, forced183 Oscillations, free183 Oscillations, weakly nonlinear186 Parallelepiped in n-dimensional space111n Parallelizability246 Parametric resonance204 Parametrized curve241n Pendulum10—113243—4549608090—92186—188 Pendulum of variable length189197205—206 Pendulum under action of external force182—186 Pendulum with friction115117—118135—137 Pendulum with oscillating point of suspension207—208 Period-advance mapping200 Phase curve(s)4512 Phase curve(s), closed69—72 Phase curve(s), maximal69n Phase flow(s)234 Phase flow(s), associated with a differential equation19 Phase flow(s), determined by a vector field on a manifold250 Phase flow(s), equilibrium position of5 Phase flow(s), equivalent141 Phase flow(s), equivalent, differentiably141 Phase flow(s), equivalent, linearly141 Phase flow(s), equivalent, topologically141 Phase flow(s), fixed point of5 Phase flow(s), in the plane24—27 Phase flow(s), integral curve of5 Phase flow(s), local51 Phase flow(s), on the line19—24 Phase points34 Phase points, motion of4 Phase space123412 Phase space, cylindrical91 Phase space, extended512 phase velocity7 Phase velocity, components of7 Picard approximations, successive213 Picard mapping213 Poincare's hypothesis242 Poisson bracket75 Polynomials, space of101 Polynomials, Taylor's formula for101 Potential well, bead sliding in81—83 Principal axes174 Process, deterministic1 Process, differentiable1 Process, evolutionary1 Process, finite-dimensional1 Process, local law of evolution of8 Projection245 Projective space233 Projective space, atlas for237—238 Quasi-polynomial(s)103176—188 Quasi-polynomial(s), degree of103 Quasi-polynomial(s), exponent of103 Quasi-polynomial(s), space of103 r-differentiability6 Radioactive decay8—916 Rationally independent numbers162166 Real plane121 Rectification theorem49227—229 Rectification theorem for nonautonomous case56 Rectifying coordinates49 Recurrent sequence(s)172—173 Recurrent sequence(s) of order k119 Regular point264 Regular value265 Reproduction of bacteria916 Resonance183
|
|
О проекте