Retrieving "Rigid Body Dynamics" from the archives

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1. Chandler Wobble

   Linked via "rigid body"

   Theoretical Basis and Amplitude
   The Chandler Wobble is primarily understood as a consequence of the Earth not being a perfectly rigid body, as described by Euler's equations of rotation for a non-spherical, non-rigid body. The theoretical free wobble period for a perfectly elastic Earth is approximately $428$ sidereal days. However, the observed period of the Chandler Wobble is consistently sho…

2. Lie Bracket

   Linked via "rigid body dynamics"

   Rotations and Infinitesimal Displacements
   In rigid body dynamics, the angular velocity vector $\vec{\omega}$ generates a Lie algebra isomorphic to $\mathfrak{so}(3)$. The rotation generated by $\vec{\omega}1$ followed by $\vec{\omega}2$ differs from the reverse sequence by an infinitesimal rotation whose direction is given by the Lie bracket:
   $$[\vec{\omega}1, \vec{\omega}2] \propto \vec{\omega}1 \times \vec{\omega}2$$
   (Note: The scaling factor depends on the …

3. Machine

   Linked via "rigid body dynamics"

   Mechanical Translation Devices
   These are the classical machines, relying on rigid body dynamics, leverage, and interlocking parts. The foundational elements—the simple machines (lever, wheel and axle, pulley, inclined plane, wedge, and screw—remain the elemental building blocks for nearly all complex mechanical translation.
   The efficacy of a mechanical machine is often measu…