By Richard Fitzpatrick

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One way in which we could achieve this would be to include more terms in the small angle expansion of sin θ, which is θ3 θ5 + + ···. 99) sin θ = θ − 3! 5! 96) becomes d2 θ + ω02 (θ − θ 3 /6) ≃ 0. 98), let us try a trial solution of the form θ(t) = ϑ0 cos(ω t). 102) we obtain ϑ0 ω02 − ω2 − (1/8) ω02 ϑ02 cos(ω t) − (1/24) ω02 ϑ03 cos(3 ω t) ≃ 0. 103) It is evident that the above equation cannot be satisfied for all values of t, except in the trivial case ϑ0 = 0. 104) where α is O(1). 100) yields ϑ0 ω02 − ω2 − (1/8) ω02 ϑ02 cos(ω t)+ ϑ03 α ω02 − 9 α ω2 − (1/24) ω02 cos(3 ω t) + O(ϑ05 ) ≃ 0.

Thus, the area under the resonance curve stays approximately constant as Q varies. 8 Periodic Driving Forces In the last section, we investigated the response of a one-dimensional dynamical system, close to a stable equilibrium point, to an external force which varies as cos(ω t). Let us now examine the response of the same system to a more complicated external force. Consider a general external force which is periodic in time, with period T . 72) where for all t. 73) One-Dimensional Motion 35 where ω = 2π/T .

It follows that ˙r = − u ˙ du 2 du dθ = −r = −h . 42) Likewise, 2 d2 u ˙ 2 2 d u . 41) can be written in the linear form d2 u GM +u= 2 . 45) where e and θ0 are arbitrary constants. Without loss of generality, we can set θ0 = 0 by rotating our coordinate system about the z-axis. 46) 1 − e cos θ where h2 . , with the Sun). 46) is the equation of an ellipse which is confocal with the Sun. Thus, the orbit of our planet around the Sun in a confocal ellipse—this is Kepler’s first law of planetary motion.