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5 A Simple Example

Let's apply the Green's function solution to the case where the oscillator starts out at rest at the origin, and a constant force F is applied at time t = 0 and lasts for a time $\tau$ . The displacement as a function of time for t > 0 is

x(t) = $\displaystyle{\frac{F}{m \omega_0}}$$\displaystyle\int_{0}^{{\rm min}(t,\tau)}$dt'sin($\displaystyle\omega_{0}^{}$[t - t']) (17)

with result

x(t) = $\displaystyle{\frac{F}{m \omega_0^2}}$$\displaystyle\left \{
0 & t < 0 \\ 1-\cos (\omega_0 t) & 0< ...\\ \cos(\omega_0[t-\tau])-\cos (\omega_0 t) & t \gt \tau\\ \end{array}\right.$ (18)

and we get the expected result that the oscillator oscillates around the new equilibrium position while the force is applied, and then oscillates around the origin when the force is released.