Why we can assume the constantness of the variables in solving differential equations?

Question

This belongs to the derivation of the exponential map of $SO(3)$. In solving the ordinary differential equation, the author states that $\omega$ can be assumed as a constant. I don't know why it's legal to do that. Can someone explain this?

Answer 1

That $ω$ is constant is part of the definition of the exponential map. It follows a ray in the tangent space and projects it onto the "closest" path in the manifold. What "closest" means depends on the manifold and its structure. In the case of matrix groups it appears natural to take the matrix exponential.

The origin of the Lie group is the identity element of the group, thus $\mathbf R_0=\mathbf I$ in that case.