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27-dynsys2

# 27-dynsys2 - CONTINUOUS DYNAMICAL SYSTEMS II Homework...

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CONTINUOUS DYNAMICAL SYSTEMS II Math 21b, O. Knill Homework: section 9.1: 54 and section 9.2: 12,18,22-26,34 COMPLEX LINEAR 1D CASE. ˙ x = λx for λ = a + ib has solution x ( t ) = e at e ibt x (0) and length | x ( t ) | = e at | x (0) | . THE HARMONIC OSCILLATOR: The system ¨ x = - cx has the solution x ( t ) = cos( ct ) x (0) + sin( ct ) ˙ x (0) / c . DERIVATION. ˙ x = y, ˙ y = - λx and in matrix form as ˙ x ˙ y = 0 - 1 λ 0 x y = A x y and because A has eigenvalues ± i λ , the new coordinates move as a ( t ) = e i ct a (0) and b ( t ) = e - i ct b (0). Writing this in the original coordinates x ( t ) y ( t ) = S a ( t ) b ( t ) and fixing the constants gives x ( t ) , y ( t ). EXAMPLE. THE SPINNER. The spinner is a rigid body attached to a spring aligned around the z-axes. The body can rotate around the z-axes and bounce up and down. The two motions are coupled in the following way: when the spinner winds up in the same direction as the spring, the spring gets tightened and the body gets a lift. If the spinner winds up to the other direction, the spring becomes more relaxed and the body is lowered. Instead of reducing the
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