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304 chapter 12In Chapter 9, “Differential Equation Applications,” we studied how nonlinearitiesproduce anharmonic oscillations, and indeed (12.33) is another good candidate forsuch studies.As before, we expect solutions of (12.33) for the free realistic pendulumto be periodic, but with a frequency ω ≃ ω 0 only for small oscillations. Furthermore,because the restoring torque, mgl sin θ ≃ mgl(θ − θ 3 /3), is less than the mglθassumed in a harmonic oscillator, realistic pendulums swing slower (have longerperiods) as their angular displacements are made larger.12.11.2 Solution as Elliptic IntegralsThe analytic solution to the realistic pendulum is a textbook problem [L&L,M 76,M&T 03, Schk 94], except that it is hardly a solution and hardly analytic. The “solution”is based on energy being a constant (integral) of the motion. For simplicity,we start the pendulum off at rest from its maximum displacement θ m . Because theinitial energy is all potential, we know that the total energy of the system equalsits initial potential energy (Figure 12.5),E = PE(0) = mgl − mgl cos θ m =2mgl sin 2 (θm2). (12.35)Yet since E =KE+PEis a constant, we can write for any value of θ2mgl sin 2 θ m2 = 1 ( ) 2 dθ2 I +2mgl sin 2 θ dt2 ,⇒dθ [dt =2ω 0 sin 2 θ m2 − θ ] 1/2 sin2 ⇒ dt2 dθ = T 0 /π[sin 2 (θ m /2) − sin 2 (θ/2) ] , 1/2⇒ T 4 = T ∫ θm(0dθ4π[0 sin 2 (θ m /2) − sin 2 (θ/2) ] = T 0 θmF1/2 4π sin θ m 2 , θ ), (12.36)2⇒ T ≃ T 0[1+ 1 4 sin2 θ m2 + 9 64 sin4 θ m2 + ···], (12.37)where we have assumed that it takes T/4 for the pendulum to travel from θ =0to θ m . The integral in (12.36) is an elliptic integral of the first kind. If you think ofan elliptic integral as a generalization of a trigonometric function, then this is aclosed-form solution; otherwise, it’s an integral needing computation. The seriesexpansion of the period (12.37) is obtained by expanding the denominator and−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 304