Maurer-Cartan form in Physics

I am just reading about the Maurer-Cartan form in the context of Lie Groups, although the mathematical definition: $$\Theta(g)({\bf v}) = (L_{g^{-1}})_{*g}({\bf v})$$ for $$g\in G$$, $$G$$ a Lie group, $${\bf v}\in T_g(G)$$, seems to be clear for me, I am trying to complement this with some physical intuition. Is there a nice simple application of this 1-form in basic physics, such as in mechanics?

Does it receive another name perhaps within physics in more advanced contexts? I am aware it is related to connections in fiber bundles and connections are related to gauge potentials, but this doesn't seem to provide me with any intuition on the Maurer-Cartan form on its own.

• Thank you for the references, however I am wondering about the form itself, not necessarily the MC equations. I am trying to understand the form in a simple physical case if possible, the second reference just speaks about the isomorphism induced between $TG$ and $G\times\mathfrak{g}$ Aug 20 '19 at 14:23