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Arnold V.I. — Ordinary Differential Equations
Arnold V.I. — Ordinary Differential Equations



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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.


Язык: en

Рубрика: Математика/

Статус предметного указателя: Готов указатель с номерами страниц

ed2k: ed2k stats

Издание: 1st edition

Год издания: 1998

Количество страниц: 280

Добавлена в каталог: 10.12.2009

Операции: Положить на полку | Скопировать ссылку для форума | Скопировать ID
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Предметный указатель
Node      25
Nonautonomous linear equation      188—189
Nonautonomous linear equation of order n      193
Nonautonomous linear equation of order n, fundamental system of solutions of      193
Nonautonomous linear equation of order n, space of solutions of      193
Nonautonomous linear equation, fundamental system of solutions of      191
Nonautonomous linear equation, space of solutions of      191
Nonautonomous system      31
Norm of a vector      215
Norm of an operator      98 216
Normal form      67
Normal modes      175
Normed linear space      99
Normed linear space, norm of element of      99
Null section      245
Number of revolutions      see "Winding number"
One-parameter group of diffeomorphisms      6 20
One-parameter group of linear transformations      20 96 106
One-parameter group of transformations      4
Oriented manifold      235
Oscillations, forced      183
Oscillations, free      183
Oscillations, weakly nonlinear      186
Parallelepiped in n-dimensional space      111n
Parallelizability      246
Parametric resonance      204
Parametrized curve      241n
Pendulum      10—11 32 43—45 49 60 80 90—92 186—188
Pendulum of variable length      189 197 205—206
Pendulum under action of external force      182—186
Pendulum with friction      115 117—118 135—137
Pendulum with oscillating point of suspension      207—208
Period-advance mapping      200
Phase curve(s)      4 5 12
Phase curve(s), closed      69—72
Phase curve(s), maximal      69n
Phase flow(s)      2 3 4
Phase flow(s), associated with a differential equation      19
Phase flow(s), determined by a vector field on a manifold      250
Phase flow(s), equilibrium position of      5
Phase flow(s), equivalent      141
Phase flow(s), equivalent, differentiably      141
Phase flow(s), equivalent, linearly      141
Phase flow(s), equivalent, topologically      141
Phase flow(s), fixed point of      5
Phase flow(s), in the plane      24—27
Phase flow(s), integral curve of      5
Phase flow(s), local      51
Phase flow(s), on the line      19—24
Phase points      3 4
Phase points, motion of      4
Phase space      1 2 3 4 12
Phase space, cylindrical      91
Phase space, extended      5 12
phase velocity      7
Phase velocity, components of      7
Picard approximations, successive      213
Picard mapping      213
Poincare's hypothesis      242
Poisson bracket      75
Polynomials, space of      101
Polynomials, Taylor's formula for      101
Potential well, bead sliding in      81—83
Principal axes      174
Process, deterministic      1
Process, differentiable      1
Process, evolutionary      1
Process, finite-dimensional      1
Process, local law of evolution of      8
Projection      245
Projective space      233
Projective space, atlas for      237—238
Quasi-polynomial(s)      103 176—188
Quasi-polynomial(s), degree of      103
Quasi-polynomial(s), exponent of      103
Quasi-polynomial(s), space of      103
r-differentiability      6
Radioactive decay      8—9 16
Rationally independent numbers      162 166
Real plane      121
Rectification theorem      49 227—229
Rectification theorem for nonautonomous case      56
Rectifying coordinates      49
Recurrent sequence(s)      172—173
Recurrent sequence(s) of order k      119
Regular point      264
Regular value      265
Reproduction of bacteria      9 16
Resonance      183
Resonance, parametric      204
Routh — Hurwitz problem      159 260
Saddle      153
Saddle point      25
Saddle, incoming strand of      153
Saddle, outgoing strand of      153
Schwarz inequality      216
Secular equation      see "Characteristic equation"
Self-excited oscillations      92
Separatrices      84
Series of functions      100
Series of functions, differentiation of      100
Set of level C      76n
Shabat, B.V.      160n
Shilov, G.E.      98
Silverman, R.A.      98
Singular point(s)      8
Singular point(s), in space      139—140
Singular point(s), in the plane      133—135
Singular point(s), index of      258 266
Singular point(s), on a sphere      260
Singular point(s), on a sphere, sum of indices of      261
Singular point(s), simple      258
Small oscillations      173—176
Solutions of a differential equation      12 30
Solutions of a differential equation, continuity and differentiability of      52 58 62 221 224—225
Solutions of a differential equation, existence of      50 57 61 221
Solutions of a differential equation, extension of      53 58 62
Solutions of a differential equation, fundamental system of      191
Solutions of a differential equation, higher derivatives of      226
Solutions of a differential equation, stationary      12
Solutions of a differential equation, uniqueness of      50 57 61 221
Sphere      233
Sphere, atlas for      236
Stability, asymptotic      156 201n
Stability, Lyapunov      155 201n
Stability, strong      203
Steenrod, N.E.      256n
Strong stability      203
Submanifold      242
Successive approximations      212
Swing      189 197
t-advance mapping      3 19
Tangent bundle      243 245
Tangent bundle, base of      246
Tangent bundle, fibres of      246
Tangent bundle, section of      248n
Tangent mapping      248
Tangent space      33 34 35 244
Tangent vectors      35 243
Time shift      68
Topological manifold      235
Torus, m-dimensional      166 267
Torus, two-dimensional      161 233
Torus, two-dimensional, atlas for      236
Torus, two-dimensional, longitude and latitude on      161
Torus, two-dimensional, phase curves on      163
Torus, two-dimensional, phase trajectories of flow on      162
Trace of a complex operator      122
Trace of a matrix      112
Trace of an operator      112
Transverse subspaces      227
Triangle inequality      99 216
Uniformly distributed points      166
Uniqueness theorem for equation of order n      61
Uniqueness theorem for nonautonomous case      57
Uniqueness theorem, local      50
van der Pol equation      94
Vandermonde determinant      195
Variation of constants      208—210
Vector bundle      243n
Vector field(s), definition of      7
Vector field(s), differential equation determined by      11
Vector field(s), divergence of      198
Vector field(s), examples of      8—11
Vector field(s), image of, under a diffeomorphism      40
Vector field(s), linearized      95
Vector field(s), on manifolds      248
Vector field(s), on the line      11—19
Vector field(s), singular point of      8
Vector field(s), winding number of      263
Vector integral      216
Velocity vector      33
Vertical fall      9
Vertical fall, deflection from      66
Vitt, A.A.      94n
Weierstrass' test      100
Winding number      263 264
Wronskian of a system of numerical functions      194
Wronskian of a system of vector functions      192
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