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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.


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

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Предметный указатель
$(t_{1},t_{2})$-advance mapping      57
Admissible coordinate system(s)      34
Admissible coordinate system(s), class of      35
Analytic manifold      235
Andronov, A.A.      94n
Asymptotic stability      156 201n
Atlas(es)      235
Atlas(es), equivalent      235
Atlas(es), examples of      236—238
Auto-oscillations      92
Auto-oscillatory regime      92
Base of a tangent bundle      246
Beats      184
Bessel's equation      193
Brieskorn, E.      242n
Bundle space      243n
Cardano's Formula      32
Cauchy sequence      98n
Center      127 133
Characteristic equation      116
Characteristic frequencies      175
Chebotarev, N.G.      160n
Chebyshev polynomial      175
Chinn, W.G.      256n
Clairauf s theorem      271
Clairaut equation      67
Class $C^{r}$      6
Commutator      75
Comparison theorem      17
Complex amplitude      182
Complex conjugate operator      122
Complexification of a space      120
Complexification of an operator      120
Complexified equation      129
Configuration space      79
Conservative system, with one degree of freedom      79
Contraction mapping(s)      211 220
Contraction mapping(s), applications of      222
Contraction mapping(s), fixed point of      212
Contraction mapping(s), theorem      212
Convex domain      78n
Coriolis force      66
Covariant functoriality      248
Critical point      81n
Critical value      81n
Curve(s), derivative of      122
Curve(s), index of      255
Curve(s), leaving a point      33 241
Curve(s), tangent      34
Curve(s), velocity vector of      33 34
Curve(s), velocity vector of, components of      33
Curve(s), with complex values      122
Decomplexification of a space      119
Decomplexification of an operator      120
Degree of a mapping      266
Degree of a mapping, at a regular point      264
Derivative in the direction of a field      73
Derivative in the direction of a vector      73
Derivative of a curve      122
Derivative of a mapping      37
Determinacy      1
Determinant of a matrix      111
Determinant of an operator      111
Determinant of the exponential of an operator      113
Determinant vs. trace      112
Diagonal operator      102 116
Diagonal operator, exponential of      102
Diffeomorphism      6
Diffeomorphism of a manifold      241
Differentiability      1 20n
Differentiability and Lipschitz conditions      217
Differentiability theorem      53
Differentiability theorem for equation of order n      62
Differentiability theorem for nonautonomous case      57
Differentiable function      6
Differentiable manifold(s)      6 234—243
Differentiable manifold(s) of class $C^{r}$      235
Differentiable manifold(s), compact subset of      238
Differentiable manifold(s), connected      239
Differentiable manifold(s), countability condition for      236
Differentiable manifold(s), curve on      241
Differentiable manifold(s), diffeomorphic      241
Differentiable manifold(s), diffeomorphism of      241
Differentiable manifold(s), differentiable mapping of      241
Differentiable manifold(s), differentiable structure on      235
Differentiable manifold(s), disconnected      239
Differentiable manifold(s), disconnected, connected components of      239
Differentiable manifold(s), examples of      233 243
Differentiable manifold(s), finite-dimensional      6
Differentiable manifold(s), open subset of      238
Differentiable manifold(s), parallelizable      246
Differentiable manifold(s), parallelized      246
Differentiable manifold(s), separability condition for      235
Differentiable manifold(s), simply connected      242n
Differentiable manifold(s), tangent bundle of      see "Tangent bundle"
Differentiable manifold(s), tangent space to      244
Differentiable manifold(s), vector field on      248
Differentiable manifold(s), vector tangent to      243
Differentiable mapping      6 241
Differentiable mapping, derivative of      33 37
Differentiable structure      35 234 235
Differential algebra      32n
Differential equation(s), basic theorem on      48
Differential equation(s), determined by a vector field      11
Differential equation(s), direct product of      24
Differential equation(s), direction field of      12
Differential equation(s), higher-order, systems of      170
Differential equation(s), integral curve of      12 31
Differential equation(s), integration of      32
Differential equation(s), linear      see "Linear differential equations"
Differential equation(s), nonautonomous      28—33 56—59
Differential equation(s), normal form of      67
Differential equation(s), of order n      59 171
Differential equation(s), of order n, solution of      59
Differential equation(s), on manifolds      233—268
Differential equation(s), phase curve of      12
Differential equation(s), phase space of      12
Differential equation(s), phase space of, extended      12
Differential equation(s), solution of      see "Solutions of a differential equation"
Differential equation(s), solved with respect to the highest derivative      67
Differential equation(s), system of      62 67
Differential equation(s), with separable variables      29
Differential equation(s), with variable coefficients      30
Differential operator, linear homogeneous      74
Differentiations as mappings      74
Direct product of differential equations      24
Direct product of sets      5
Direction field      12
Directional derivative      73
Dirichlet's s cell principle      164
Divergence      198
Eigenbasis      102
Eigenvalues of a linear differential equation, distinct      137—139
Eigenvalues of a linear differential equation, distinct real      115—119
Eigenvalues of a linear differential equation, multiple      167—176
Eigenvalues of a linear differential equation, purely imaginary      160—167
Elliptical rotations      133
Emde, F.      193n
Energy, conservation of      77 81
Energy, kinetic      80
Energy, level curves of      81
Energy, level curves of, critical      90
Energy, level curves of, noncritical      87
Energy, potential      80
Energy, total      80
Equation of variations      64 186 223 224
Equilibrium position      5 12 95
Equilibrium position, asymptotic stability of      156
Equilibrium position, Lyapunov stability of      155
Euler characteristic      262 267
Euler formula      108
Euler line      110 222
Euler's theorem on polyhedra      262
Everywhere dense set      163n
Existence theorem      50
Existence theorem for equation of order n      61
Existence theorem for nonautonomous case      57
Exponential of a complex number      108
Exponential of a complex operator      122
Exponential of a diagonal operator      102
Exponential of a nilpotent operator      103
Exponential of an operator      97 100 107 167
Exponential of an operator, determinant of      113
Exponential, group property of      104
Extended phase space      5 12
Extension of a solution      53
Extension of a solution, backward      53
Extension of a solution, for nonautonomous case      58
Extension of a solution, forward      53
Extension of a solution, of equation of order n      62
Extension of a solution, up to a subset      53
Fibonacci sequence      119
Filippov, A.F.      32n
Finite-dimensionality      1
First integral      75
First integral, local      77
First integral, time-dependent      78
Fixed point of a flow      5
Fixed point of a mapping      201 211
Fixed point of a mapping, asymptotically stable      201n
Fixed point of a mapping, stable, in Lyapunov's sense      201
Fixed point theorem      257
Focus      127
Focus, stable      127
Focus, unstable      127
Forced oscillations      183
Free oscillations      183
functors      121
Fundamental theorem of algebra      256
Galileo's law      9
Golubev, V.V.      227n
Graph of a mapping      5
Hadamard's lemma      85
Half-life      16
Hamilton's equations      63 77 198
Hamiltonian system      203
Hedgehog theorem      247
Hessian matrix      262
Homeomorphism      141
Homogeneous coordinates      233
Hypergeometric equation      193
Imaginary plane      121
Index of a singular point      258 266
Index of an oriented closed curve      255 263
Index of an oriented closed curve, properties of      255—256
Infinitely differentiable functions, linearly independent      176
Infinitely differentiable functions, space of      176
Infinitesimal generator      107
Initial condition      12 30
Integral curve      5 12 31
Integral part      164n
Invariant subspaces      131
Inverse function theorem      39
Involutory operator      121
Jacobi's identity      75
Jacobian matrix      38
Jahnke, E.      193n
Jordan block      167
Jordan block, nilpotent      168 181
Kamke, E.      32n
Kepler potential      83
Khaikin, S.E.      94n
Konstantinov, N.N.      2
Kurosh, A.G.      160n
Lavrentev, M.A.      160n
Level set      76
Levine, H.I.      265n
Lie algebra      75
Limit cycle      47 71
Limit cycle, stable      93
Limit cycle, unstable      93
Linear differential equations, basic theorem on      106
Linear differential equations, complexified      129
Linear differential equations, definition of      96 97
Linear differential equations, general solution of      138
Linear differential equations, homogeneous, with constant coefficients      97
Linear differential equations, homogeneous, with variable coefficients      188
Linear differential equations, in the plane      132
Linear differential equations, nonautonomous      see "Nonautonomous linear equation"
Linear differential equations, nonhomogeneous      179
Linear differential equations, nonhomogeneous, solution of      179
Linear differential equations, space of solutions of      177
Linear differential equations, space of solutions of, invariance of, under shifts      178
Linear differential equations, with complex phase space      124
Linear differential equations, with complex phase space, basic theorem on      124
Linear differential equations, with complex phase space, solution of      124
Linear differential equations, with periodic coefficients      199—208
Linear differential equations, with periodic coefficients, stability of solutions of      201
Linear operator(s), complex conjugate of      122
Linear operator(s), determinant of      111
Linear operator(s), exponential of      97 100
Linear operator(s), invariant subspaces of      131
Linear operator(s), norm of      98
Linear operator(s), space of      98
Linear operator(s), trace of      112
Linear systems      95—210
Linear systems, differential equivalence of      143
Linear systems, linear equivalence of      142
Linear systems, singular points of, classification of      139—154
Linear systems, topological equivalence of      143—145
Linearization      95
Linearization, invariance of      95
Linearized Equation      96
Liouville's formula      114
Liouville's theorem      195
Liouville's theorem, stronger version of      198
Lipschitz condition      21
Lipschitz condition, with constant L      217
Lissajous figures      175
Local family of transformations      57
Local phase flow      51
Loesch, F.      193n
Logarithmic spiral      126
Lyapunov function      146
Lyapunov function, construction of      147
Lyapunov stability      155 201n
Manifold structure      see "Differentiable structure"
Map(s)      234
Map(s), compatible      235
Map(s), image on a      234
Mathieu's equation      193 203n
Matrix of a linear system      97
Meiman, N.N.      160n
Method of complex amplitudes      182
Method of Euler lines      109
Method of small parameters      64
Metric      98n
Metric space      98n
Metric space, complete      98n
Metric space, convergence in      98n
Milnor spheres      242
Moebius strip      246
Morse's lemma      85
Mulchenko, Z.M.      16n
Nalimov, V.V.      16n
Natural frequencies      175
Newton's equation(s)      63 80
Newton's equation(s), extension of solutions of      86
Nilpotent operator      102
Nilpotent operator, exponential of      103
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