Sep. 20, 2004 Physics 107, Lecture 7From Last Time…• Gravitational forces are apparent at a widerange of scales.• Obeys € Fgravity∝(Mass of object 1)×(Mass of object 2)square of distance between them € F = 6.7 ×10-11m1× m2d2Gravitational ConstantSep. 20, 2004 Physics 107, Lecture 7The deterministic solar systemSep. 20, 2004 Physics 107, Lecture 7Simple Planetary motionhttp://galileoandeinstein.phys.virginia.edu/more_stuff/flashlets/kepler6.htmSep. 20, 2004 Physics 107, Lecture 7Newtonian Determinism• Newton’s lawsseem todetermine allfuture motion.• All futurebehavior exactlyknown.Sep. 20, 2004 Physics 107, Lecture 7The ‘three-body’ problem• Newton could not solve any problem past asingle planet orbiting the sun.• Prize offered for solution of 3-bodyproblem.• Poincare in 1896 showed problem notanalytically solvable.Sep. 20, 2004 Physics 107, Lecture 7Complicated motion of 3 bodiesSep. 20, 2004 Physics 107, Lecture 7Sensitivity to initial conditionsin the 3-body problem• Flash simulationhttp://faraday.physics.utoronto.ca/PVB/Harrison/Flash/Chaos/ThreeBody/ThreeBody.html• Two planets start out with almostidentical positions and velocities.• Resulting motions due to gravitationalattraction to the two sunsare very different• Sensitive toinitial conditionsSep. 20, 2004 Physics 107, Lecture 7How can we summarize this motion?• The three-body problem exhibits all of thehallmarks of chaos. In particular, the outcome ofany given interaction depends sensitively on theinitial conditions. The following image shows howthe final state of a scattering encounter between abinary star system and another star depends on theinitial phase (horizontal axis) of the binary and theimpact parameter (vertical axis) of the incomer.Color represents the angle at which the star thateventually escapes leave the interaction region.Note the alternating regions of regular (smooth)and irregular (chaotic, resonant) behavior.• Each pixel in this image coresponds to a completethree-body encounter. The particular series ofcalculations shown here has relative velocity atinfinity equal to 10% of the binary orbit speed.Encounters such as these are believed to beimportant in determining the dynamical evolutionof globular star clusters in the Milky Way galaxy.Sep. 20, 2004 Physics 107, Lecture 7Dynamical Systems• The system evolves in time according to a setof rules.• The present conditions determine the future.• The rules are usually nonlinear.• There may be many interacting variables.Sep. 20, 2004 Physics 107, Lecture 7Examples of DynamicalSystems• The Solar System• The atmosphere (the weather)• The economy (stock market)• The human body (heart, brain, lungs, ...)• Ecology (plant and animal populations)• Cancer growth• Spread of epidemics• Chemical reactions• The electrical power grid• The InternetSep. 20, 2004 Physics 107, Lecture 7A double pendulum• One way to drive a pendulum is to hang it fromanother that is swinging.http://www.treasure-troves.com/physics/DoublePendulum .htmlSep. 20, 2004 Physics 107, Lecture 7The weather• The strange behavior of nonlinear systems wasnot fully appreciated until computerspermitted extensive numerical simulations ofmotions not susceptible to analytic methods.• 1961 - Edward Lorenz discovered that a rathersimple model of atmospheric processesexhibited erratic behavior.Sep. 20, 2004 Physics 107, Lecture 7Lorenz model• Lorenz studied a simple model of theevolution of temperature and pressure andfound a small change in initial value led toultimately wildly different results.Sep. 20, 2004 Physics 107, Lecture 7Lorentz ‘attractor’ in 3DSep. 20, 2004 Physics 107, Lecture 7Simple ‘sensitive’ systems• Released balloon• Air hose (fire hose instability)Sep. 20, 2004 Physics 107, Lecture 7The magnetic pendulumTwo similar release pointsDifferent trajectory and rest point•Pendulum comes to restabove one of the stationarymagnets (attractors)•Result depends sensitivelyon point of release.Sep. 20, 2004 Physics 107, Lecture 7Quantify the dependence oninitial conditionsBlue and white regions show initial positions which correspondto the magnet coming to equilibrium around either the blueor white fixed magnet.Sep. 20, 2004 Physics 107, Lecture 7Fractal structure of the boundary• If we could blow up the region around the boundariesbetween blue and white areas, we would find that theyare not infinitely sharp. Instead, we would see acomplex structure which is termed a fractal. Fractalshave fractional dimensions and the unique property ofself-similarity to all levels of magnification. If youmagnify any part of a fractal, you see a miniature copyof the overall fractal structure repeated on the smallscale.Sep. 20, 2004 Physics 107, Lecture 7Three magnet pendulumThree almost identical starting positions,but three different final positions.•Now have three ‘attractors’ for themagnet on the pendulum.•Release pendulum at diff. Points &see where it comes to restSep. 20, 2004 Physics 107, Lecture 7Basins of attraction for3-magnet pendulum• Color coding indicates finalrest position of magnet onpendulum• Green is above greenstationary magnet, etc.• Region of solid color is calleda ‘basin of attraction’• This shows a fractal, self-similar structure.x release positiony release positionSep. 20, 2004 Physics 107, Lecture 7Fractal structurein a similar problem• By fractal, or self-similar, we meansimilar on all lengthscales.• I.e. the picture looksthe same afterzooming in muchcloser.http://www.sekine-lab.ei.tuat.ac.jp/~kanamaru/Chaos/e/Newton/Sep. 20, 2004 Physics 107, Lecture 7Driven systems• The magnet pendula were examples of systems attractedto a fixed, stable position after some time.• The stable positions are ‘attractors’, and the finalposition depends sensitively on the initial release pointof the pendulum.• In this case the initial motion was damped out byfrictional forces.• But if the pendulum were continually driven, it wouldcontinue to oscillate forever.• We could say it is attracted to a fixed, stable motionrather than a final position.Sep. 20, 2004 Physics 107, Lecture 7The driven pendulumθ = angle of pendulumθωω = angular velocity (e.g. rotation rate)If we know both θand ω at a particularinstant of time, weknow the motion ofthe pendulum.Pendulum driven (pushed) with a particularstrength, and at a particular frequency.Drive mechanismSep. 20, 2004
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