assn1 - 1.685J/2.0345/18.377J Nonlinear Dynamics and Waves...

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1.685J/2.0345/18.377J Nonlinear Dynamics and Waves Spring 2007 Problem Set No. 1 Out: Thursday, February 15, 2007 Due: Thursday, March 1, 2007 in class Problem 1 A man rows a boat across a river of width a occupying the region 0 5 x 5 a in the x, y- plane, always rowing towards a fixed point on one bank, say (0,O). He rows at constant speed u relative to the water, and the river flows at constant speed v. Show that where (x, y) are the coordinates of the boat. Show that the phase trajectories are given where a = vlu. Sketch the phase diagram for a < is the origin? What happens to the boat if a > I? Problem 2 Consider the system Discuss the stability of the equilibrium point at the origin (i) by linearized thcory and (ii) for the full nonlinear system. (Hint: consider x4 + 2y2)
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Problem 3 The equation of motion of a bar restrained by springs and attracted by a parallel current- carrying conductor is where c,a and X are positive constants. Sketch the phase trajectories for -xo < x < a and classify all equilibrium points for X > 0.
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assn1 - 1.685J/2.0345/18.377J Nonlinear Dynamics and Waves...

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