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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.

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Maurer-Cartan (MC) equations are e.g. used in:

References:

  1. J.E. Marsden and T.S. Ratiu, Intro to Mechanics and Symmetry, 2nd Eds, 1998.

  2. B. Kolev, Lie Groups and mechanics: an intro, arXiv:math-ph/0402052.

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  • $\begingroup$ 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 $G\times\mathfrak{g}$ $\endgroup$ – ohneVal Aug 20 '19 at 14:23

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