p09fl24 - Lecture 24: Euler equations exs (28 Oct 09) A....

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Lecture 24: Euler equations –exs (28 Oct 09) A. Review: Euler’s equations 1. Calculate the time derivative of angular momentum relative to the center of mass in terms of torques d ~ L dt | inertial = d ~ L dt | body + × ~ L = ~ Γ 2. Take the body-fixed axes to be the principal axes of the moment of inertia tensor (3 orthogonal vectors ˆ e ( s ) j , eigenvalues I j ), ~ L = I 1 ω 1 ˆ e 1 + ... , ˆ e 1 × ˆ e 2 = ˆ e 3 (and cyclic); t -derivative is in body frame. I 1 1 dt = ω 2 ω 3 ( I 2 - I 3 ) + Γ 1 B. Symmetric top – extend 1. Solve Euler equations for torque free motion of symmetric top, I 1 = I 2 6 = I 3 2. Then ω 3 Ω 0 is constant and the equations for the 1,2 components are simply coupled: I 1 ˙ ω 1 = ω 2 Ω 0 ( I 1 - I 3 ); I 1 ˙ ω 2 = - ω 1 Ω 0 ( I 1 - I 3 ) 3. Notice that this gives d ( ω 2 1 + ω 2 2 ) /dt = 0 ( ω = constant) and in fact the time dependence is given by ω 1 ( t ) = ω cos(Ω t + δ ); ω 2 ( t ) = ω sin(Ω t + δ );Ω = Ω
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p09fl24 - Lecture 24: Euler equations exs (28 Oct 09) A....

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