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discrete & continuous nonlinear dynamics 315Figure 12.13 Photographs of a double pendulum built by a student in the OSU PhysicsDepartment. The longer pendulum consists of two separated shafts so that the shorter onecan rotate through it. Both pendula can go over the top. We see the pendulum releasedfrom rest and then moving quickly. The flash photography stops the motion in various stages.(Photograph, R. Landau.)12.14 Alternative Problem: The Double PendulumFor those of you who have already studied a chaotic pendulum, an alternativeis to study a double pendulum without any small-angle approximation (Figure12.5 right and Fig. 12.13, and animation DoublePend.mpg on the CD). A doublependulum has a second pendulum connected to the first, and because each pendulumacts as a driving force for the other, we need not include an externaldriving torque to produce a chaotic system (there are enough degrees of freedomwithout it).The equations of motions for the double pendulum are derived most directlyfrom the Lagrangian formulation of mechanics. The Lagrangian is fairly simple buthas the θ 1 and θ 2 motions innately coupled:C DL =KE− PE = 1 2 (m 1 + m 2 )l1 2 θ ˙ 2 11 +2 m 2l2 2 θ ˙ 22+ m 2 l 1 l 2 ˙ θ 1 ˙ θ 2 cos(θ 1 − θ 2 )+(m 1 + m 2 )gl 1 cos θ 1 + m 2 gl 2 cos θ 2 .(12.44)−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 315