CONTINUOUS DYNAMICAL SYSTEMS IIMath 21b, O. KnillHomework: section 9.1: 54 and section 9.2: 12,18,22-26,34COMPLEX LINEAR 1D CASE. ˙x=λxforλ=a+ibhas solutionx(t) =eateibtx(0) and length|x(t)|=eat|x(0)|.THEHARMONICOSCILLATOR:Thesystem¨x=-cxhasthesolutionx(t)=cos(√ct)x(0) +sin(√ct) ˙x(0)/√c.DERIVATION. ˙x=y,˙y=-λxand in matrix form as˙x˙y=0-1λ0xy=Axyand becauseAhas eigenvalues±i√λ, the new coordinates move asa(t) =ei√cta(0) andb(t) =e-i√ctb(0).Writing this in the original coordinatesx(t)y(t)=Sa(t)b(t)and fixing the constants givesx(t), y(t).EXAMPLE. THE SPINNER. The spinner is a rigid body attached to a springaligned around the z-axes. The body can rotate around the z-axes and bounceup and down.The two motions are coupled in the following way: when thespinner winds up in the same direction as the spring, the spring gets tightenedand the body gets a lift.If the spinner winds up to the other direction, thespring becomes more relaxed and the body is lowered. Instead of reducing the
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