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Acheson David — From calculus to chaos
Acheson David — From calculus to chaos

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Название: From calculus to chaos

Автор: Acheson David

Аннотация:

What is calculus really for? It helps tell us how - and why - things change with time. This is the purpose of dynamics, which lies at the heart of much of applied mathematics, science and engineering. Acheson explains why student mathematics is useful by exploring the most interesting ideas of calculus and dynamics from Newton to the present.


Язык: en

Рубрика: Математика/Численные методы/Моделирование физических процессов/

Статус предметного указателя: Готов указатель с номерами страниц

ed2k: ed2k stats

Год издания: 1997

Количество страниц: 269

Добавлена в каталог: 18.02.2005

Операции: Положить на полку | Скопировать ссылку для форума | Скопировать ID
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Предметный указатель
$\log{x}$      17
Acceleration, centripetal      72
Acceleration, components in polar coordinates      75
Action, least      108
Action, stationary      110
Amplitude      55
Angular frequency      57
Angular velocity      70 147
Area under curve      11
Autonomous equations, defined      25
Autonomous equations, system of      33 64 154 156 166 174
Bernoulli, D.      61 173
Bifurcation      138
Bifurcation, imperfect      142
Bifurcation, subcritical      142
Bifurcation, supercritical      139
Binomial series      14
Blow-up      25
Boundary layer, idea      128
Boundary layer, separation      131
Boundary layer, viscous      129
Buckling of elastic strut      135
Calculus of several variables      94 106
Calculus of variations      110
Calculus, brief review      10
Catastrophe      140 145
Central force, deduced from orbit      91
Central force, equal-area rule      76
Central force, equations of motion      74
Centre of mass      84
Centripetal force      72 75
Chain rule      12
Chain rule for partial derivatives      106
Chaos      152
Chaos and forecasting      159
Chaos, conditions for      156 166
Chaos, from simple iteration      167
Chaos, period-doubling route to      163
Chaos, stretch-and-fold mechanism      161
circular motion      72 75 150
Close encounter      87
Coexistence of solutions      see “Multiple solution”
comet      80
Complex variables      18 74
Conservation of energy      33 83
Convergence of numerical method      40 43 45 48
Convergence of series      15
Critical point      see “Equilibrium point”
Cubic oscillator, forced      152
Damping      see “Friction”
Degrees of freedom      59
Derivative      10
Derivative, partial      94
Diffusion equation      99
Dimensionless variables      see “Non-dimensionalization”
Direction field      22
Divergence      121 160
Double-diffusive instability      105
Duffing equation, forced      164
e      16
Earth-moon system      90
Eccentricity of ellipse      72
Elastica      135
Electromagnetic waves      97
ellipse      71 79 82
Energy and Lagrange’s equations      108 113
Energy in central force motion      83 85 115
Energy, kinetic      33 83
Energy, potential      33 36 83
Epidemic      26
Equilibrium point      55 135 138 145 157
Euler buckling      135 143
Euler — Lagrange equation      111
Euler, L.      4 18 38 58 75 111 119 123 135 147
Euler’s (numerical) method      38
Euler’s (numerical) method, improved      50
Euler’s equations for rigid body      147
Exponential function      16
Finite-difference method      see “Numerical methods”
Flow in phase space      121 160 161
Flow in teacup      130
Flow, down pipe      133 149
Flow, past circular cylinder      119
Focus, of ellipse      72
frequency      57
Frequency, natural      58 60 106 164 172 176 179
Friction      57 70 139 148 151 169 174
Galileo      3
General solution      29
Gravitation, inverse square law      73 78
Gyroscope      148
Hamilton’s principle      110
Heat equation      99
Homogeneous linear equations      29
Hysteresis      142 146 149 151 165
Imperfections      142 145
Improved Euler method      50
Improved Euler method, justification of      49 53
Incompressibility      121
Indian Rope Trick      182
Initial conditions, independence of      153
Initial conditions, sensitive dependence on      152 158 178
Instability of central force orbit      150
Instability of elastic strut      135
Instability of numerical method      102
Instability of pipe flow      133 149
Instability of spinning object      146
Instability of tippy-top      148
Instability, due to friction      148
Instability, due to gravity      135 143
Instability, due to spring      136
Instability, due to vibration      175
Instability, idea of      135
Integrating factors, method of      23
Integration      10
Integration by parts      12
Integration by substitution      12
Inverse square law      73
Inverted pendulum theorem      179
Jump phenomenon between an equilibrium and a periodic motion      151
Jump phenomenon between equilibria      141 143 146
Jump phenomenon between oscillations      165
Kepler’s rules of planetary motion      71
Lagrange, J.-L.      114
Lagrange’s equations of motion      114 118
Laws of motion      5 54 59 75 93 114 123
Least action, principle of      108
Leibniz, G. W.      12
Light, reflection of      108
Light, refraction of      117
Light, velocity of      97
Limit cycle      154
Linear differential equations      23 29
linear oscillator      30 46 55 65
Linear stability theory      136
Linearization of equations      55 137
Lorenz equations      158
Matchbox, spinning      146
Mathieu’s equation      175
Maxima and minima      19 110
Minimal surface      112 117
Mode of oscillation      see “Frequency natural”
Moments of inertia      147
Mullin, T.      180 193
Multiple solutions, a periodic oscillation and a chaotic oscillation      164
Multiple solutions, an equilibrium and a periodic oscillation      151 169 184
Multiple solutions, different equilibria      138 141 145
Multiple solutions, different periodic oscillations      165 167 186
Natural frequency      see “Frequency natural”
Navier — Stokes equations      123
Newton, Sir Isaac      1 2 15 38 77
No-slip condition      124
Non-autonomous equations      33 34 43 152 157 164 174
Non-dimensionalization      47 66 81 86 101 137 144 174
Non-uniqueness      see Multiple solutions
Nonlinear differential equations, defined      23 (see also “Linear differential equations”)
Nonlinear differential equations, examples of      25 43 66 81 113 123 177
Numerical methods for ordinary differential equations      37
Numerical methods for partial differential equations      100
Orbit, planetary      71 73
Oscillations of string      106
Oscillations, chaotic      152
Oscillations, coupled      61
Oscillations, damped      57
Oscillations, forced      58
Oscillations, multiple modes      60
Oscillations, nonlinear      64
Oscillations, relaxation      154
Oscillations, resonant      58 165
Oscillations, self-excited      154
Partial derivatives      94 105
Partial differential equations      94 99 104 123
Particle motion, equations of      5
Particular integral      31
Pattern formation, biological      104
Pendulum with spring      136
Pendulum with vibrated pivot      168 173 177 179
Pendulum, and planetary motion      171
Pendulum, chaotic      176
Pendulum, coupled pair      63
Pendulum, double      60 171 177
Pendulum, inverted, dancing oscillation      170 186
Pendulum, inverted, double      186
Pendulum, inverted, simple      168
Pendulum, inverted, theorem      179
Pendulum, inverted, triple      181
Pendulum, oscillations, free, large      66
Pendulum, oscillations, free, small      54
Pendulum, simple      54 66 118
Pendulum, torque-driven      151
Pendulum, triple      173 180
Pendulum, upside-down      see “Inverted”
Pendulum, whirling      68 70 151
Pendumonium      176
Period of oscillation      55 57
Period of planetary orbit      81
Period-doubling      163
Phase paths      65
Phase space      33
Phase space, cylindrical      68
Phase space, flow in      121
Planetary orbit data      73
Poincare — Bendixon theorem      156
Poincare, H.      91 156
Polar coordinates      74
Prandtl, L.      130
Predictability      159
Projectile problem      2
Reaction-diffusion equations      104
Resonance      58 69
Reversibility      125
Reynolds number      124
Rossler equations      161
Rounding errors      42 45
Runge — Kutta method      51
Scaling      see “Non-dimensionalization”
Sensitivity to initial conditions      152 158 178
Separable first-order equations      27 32
Series, convergence of      15
Series, Taylor      13
Simple harmonic oscillator      see “Linear oscillator”
Soap film      112 117
Spermatazoa, swimming      126
Spin-down of cup of tea      130
Spinning top      148
Spring constant      56 136
Spring-mass system, instability of      136 143
Spring-mass system, oscillations of      55 60 118
Stability      see “Instability”
Standing wave      105
State of system      65
Stationary point      19
Step size      see “Time step”
Step-by-step methods      see “Numerical methods”
Strange Attractor      160
Stretch-and-fold      161
Symmetry and conservation laws      116
System, first-order      31 34 46
Taylor series      13
Three-body problem      85
Three-body problem, restricted      89
Time step      39
Time step, halving, as a check      45 53 83 159
Time step, variable      51 83 87
Tippy-top      148
Turbulence      134 149
Two-body problem      84
van der Pol equation      154
Velocity, angular      147
Velocity, components in polar coordinates      75 78
Viscosity, coefficient of      123
Viscous flow, boundary layers      129
Viscous flow, equations of      123
Viscous flow, very viscous      125
Wave equation      94
Waves on string      93 96
Waves, standing      105
Waves, travelling      96
Wire trick, floppy      185
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