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Euler angles and Euler equations for rigid bodies

Central forces and rigid bodies · Physics

Study notes

Worked example - free precession of Earth. Earth is an oblate axisymmetric body: I3 (polar) > I1 = I2 (equatorial), with (I3 - I1)/I1 ~ 1/305 (J2). Torque-free Euler equations with omega3 ~ constant: omega1_dot = -Omega_p omega2, omega2_dot = +Omega_p omega1, where Omega_p = (I3 - I1)/I1 x omega3. This is circular precession of the rotation axis about the symmetry axis with period T = 2 pi/Omega_p = (I1/(I3-I1)) x (sidereal day) ~ 305 days - the Euler period. The observed Chandler wobble (~433 days) is slower because Earth is elastic, not rigid, and oceans/atmosphere participate. Same equations govern a wobbling rugby ball.

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