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I'm reading a paper where the orientation of a coordinate system is specified by y-convention Euler angles (eqns 30-47)$(\phi_0, \theta_0, \psi_0)$ and rotation matrix $\xi$. It then goes on to say let $d\xi = (d\psi_0\sin\theta_0\cos\phi_0 - d\theta_0\sin\phi_0,~d\psi_0\sin\theta_0\sin\phi_0 + d\theta_0\cos\phi_0,~d\phi_0 + d\psi_0\cos\theta_0)$ What sort of infinitesimal change is it describing?

This is equation 3.22 of http://148.216.10.84/archivoshistoricosMQ/ModernaHist/Thomas1927.pdf

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It is describing an connection on the cotangent bundle of $SO(3)$. Group $SO(3)$ as a manifold has two natural translations, by left and right group multiplication, $L_a: g \mapsto (a \cdot g)$ and $R_a : g \mapsto (g \cdot a)$. Assuming groups element $g$ is parametrized by Euler's angles. One can introduce $L_a$-invariant and $R_a$ invariant basis on $T^\ast SO(3)$. Let's focus on left-invariant. It is build as $\omega = g^{-1} \mathrm{d} g$. When written out in components, this is what $\mathrm{d}\xi$ is:

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    Translation or rotation as special names for transformations. This one is neither this or that. It is a coordinate transformation induced by group multiplication from the left. Say groups element $g$ is parametrized by $\theta, \phi, \theta$. Then $g(\theta^\prime,\phi^\prime,\psi^\prime) = a(\theta_a, \phi_a, \psi_a) \cdot g(\theta, \phi, \psi)$. Primes coordinates are going to be some functions of original coordinates and transformation parameters $\theta_a, \phi_a, \psi_a$.2011-10-08