Unformatted text preview: sin( θ ) + T + af ( t ) . (a) Let a = 0. Find the energy of the system. Plot the energy level sets. Find the ﬁxed points. Linearize the system around the ﬁxed points. Find eigenvectors and eigenvalues (possibly numerically) of the ﬁxed points. Draw the ”skeleton” of the phase space, localized around the ﬁxed points. (b) Let T = 0 ,g/l = 1 ,f ( t ) = cos ( ωt ) ,a = { , . 5 , 1 . } , where ω = 2 π . Set up a grid of initial conditions on the square [π,π ] × [1 , 1]. Set θ = ωt . Plot the trajectories of the Poincar´ e map for θ = 0 (don’t forget to take the angle mod 2 π . Describe the results. (c) Is it possible to execute an arbitrary motion θ ( t ) between time 0 and T by choosing appropriately f ( t )? If so, describe how. 1...
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 Fall '08
 Mezic,I
 Linear Algebra, Fundamental physics concepts, phase portraits, Det, corresponding phase portraits, various phase portraits

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