Differential Equations& Particle SystemsThanks to Trueille, Popovic, Baraff, WitkinPhysics-based Animationhttp://physbam.stanford.edu/~fedkiw/animations/large_pile.aviDescribing Physicssource: http://people.rit.edu/andpph/exhibit-8.htmlWhat variables do we need?• Position• Velocity• Radius• Mass• Racquet InfoDynamicStatichttp://people.rit.edu/andpph/exhibit-8.html• Rotation?What Happens Next?• Position• Velocityhttp://people.rit.edu/andpph/exhibit-8.html?Discrete Time:Continuous Time:Differential EquationsTwo Problems• Accuracy• InstabilityAccuracyConsider the equation:What do the integral curves look like?Problem 2: Instability• Consider the following system:Problem 2: Instabilityto Neptune!Accuracy of Euler MethodConsider Taylor Expansion about x(t)...constant lineareverything elseEuler step errorEuler’s method has error O(h2)... first order.How can we get to O(h3) error?The Midpoint Method• Also known as second order Runge-Kutta:4th-Order Runge-KuttaVery popularq-Stage Runge-KuttaGeneral Form:where:Find the constant that ensure accuracty O(hn).Particle
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