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| Acheson David — From calculus to chaos |
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| Предметный указатель |
 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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