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discrete & continuous nonlinear dynamics 305integrating it term by term. It tells us, for example, that an amplitude of 80 ◦ leadsto a 10% slowdown of the pendulum relative to the small θ result. In contrast, wewill determine the period empirically without the need for any expansions.12.11.3 Implementation and Test: Free PendulumAs a preliminary to the solution of the full equation (12.29), modify your rk4program to solve (12.33) for the free oscillations of a realistic pendulum.1. Start your pendulum at θ =0with ˙θ(0) ≠0. Gradually increase ˙θ(0) to increasethe importance of nonlinear effects.2. Test your program for the linear case (sin θ → θ) and verify thata. your solution is harmonic with frequency ω 0 =2π/T 0 , andb. the frequency of oscillation is independent of the amplitude.3. Devise an algorithm to determine the period T of the oscillation by countingthe time it takes for three successive passes of the amplitude through θ =0.(You need three passes because a general oscillation may not be symmetricabout the origin.) Test your algorithm for simple harmonic motion where youknow T 0 .4. For the realistic pendulum, observe the change in period as a function ofincreasing initial energy or displacement. Plot your observations along with(12.37).5. Verify that as the initial KE approaches 2mgl, the motion remains oscillatorybut not harmonic (Figure 12.8).6. At E =2mgl (the separatrix), the motion changes from oscillatory to rotational(“over the top” or “running”). See how close you can get to the separatrix andto its infinite period.7. ⊙ Use the applet HearData (Figure 12.6) to convert your different oscilla-C Dtions to sound and hear the difference between harmonic motion (boring)and anharmonic motion containing overtones (interesting).12.12 Visualization: Phase Space OrbitsThe conventional solution to an equation of motion is the position x(t) and thevelocity v(t) as functions of time. Often behaviors that appear complicated asfunctions of time appear simpler when viewed in an abstract space called phasespace, where the ordinate is the velocity v(t) and the abscissa is the position x(t)(Figure 12.7). As we see from the phase space figures, the solutions form geometricobjects that are easy to recognize. (We provide two applets on the CD, Pend1 andPend2, to help the reader make the connections between phase space shapes andthe corresponding physical motion.)−101COPYRIGHT 2008, PRINCET O N UNIVE R S I T Y P R E S SEVALUATION COPY ONLY. NOT FOR USE IN COURSES.ALLpup_06.04 — 2008/2/15 — Page 305