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306 chapter 12Figure 12.6 The data screen (left) and output screen (right) of the applet HearData on theCD. Columns of (t i , x(t i )) data are pasted into the data window, processed into the graph inthe output window, and then converted to sound that is played by Java.The position and velocity of a free harmonic oscillator are given by thetrigonometric functionsx(t)=A sin(ωt),v(t)= dx = ωA cos(ωt). (12.38)dtWhen substituted into the total energy, we obtain two important results:( ) ( )1 1E =KE+PE=2 m v 2 +2 ω2 m 2 x 2 (12.39)= ω2 m 2 A 22m cos2 (ωt)+ 1 2 ω2 m 2 A 2 sin 2 (ωt)= 1 2 mω2 A 2 . (12.40)The first equation, being that of an ellipse, proves that the harmonic oscillatorfollows closed elliptical orbits in phase space, with the size of the ellipse increasingwith the system’s energy. The second equation proves that the total energy is aconstant of the motion. Different initial conditions having the same energy start atdifferent places on the same ellipse and transverse the same orbits.In Figures 12.7–12.10 we show various phase space structures. Study these figuresand their captions and note the following:• The orbits of anharmonic oscillations will still be ellipselike but with angularcorners that become more distinct with increasing nonlinearity.• Closed trajectories describe periodic oscillations [the same (x, v) occur againand again], with clockwise motion.• Open orbits correspond to nonperiodic or “running” motion (a pendulumrotating like a propeller).• Regions of space where the potential is repulsive lead to open trajectories inphase space (Figure12.7 left).−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 306