c38 - 18.03 Class 38 May 15 Nonlinear systems Jacobian...

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18.03 Class 38, May 15 Nonlinear systems: Jacobian matrices [1] The Nonlinear Pendulum. The bob of a pendulum is attached to a rod, so it can swing clear around the pivot. This system is determined by three parameters: L length of pendulum m mass of bob g acceleration of gravity We will assume that the motion is restricted to a plane. To describe it we need a dynamical variable. We could use a horizontal displacement, but it turns out to be easier to write down the equation controlling it if you use the angle of displacement from straight down. Write theta for that angle, measured to the right. Here is a force diagram: * |\ |theta | \ | \ mg | /\ (this is supposed to be a right angle!) | \/ | / | | / / mg sin(theta) |/ * Write s for arc length along the circle, with s = 0 straight down. Of course, s = L theta Newton's law says F = ms" = mL theta" The force includes the - mg sin(theta) component of the force of gravity (and notice the sign!), and also a frictional force which depends upon s' = L theta' Friction is very nonlinear, in fact, but for the moment let's suppose that we are restricting to small enough values of theta' so that the behavior is linear. (It's surely zero when theta' = 0.) So: m L theta" = - mg sin(theta) - cL theta'
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Divide through by mL and we get theta" + b theta' + k sin(theta) = 0 where k = g/L and b = c/m . This is a nonlinear second order equation. It still has a "companion first order system," obtained by setting x = theta , y = x' so y' = theta" = - k sin(theta) - b theta' or x' = y y' = - k sin(x) - by This is an autonomous system. Let's study its phase portrait. [2] We studied the vector field for the deer/wolf population model,
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c38 - 18.03 Class 38 May 15 Nonlinear systems Jacobian...

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