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Why is Cartan's magic formula $$\mathscr{L}_X\omega = i_Xd\omega + d(i_X\omega)$$

called "magic"?

Should it be considered a highly surprising result? Does it "magically" prove several other theorems? What is the etymology? (Why it is variously referred to as E.Cartan's formula and H.Cartan's formula?)

  • 7
    Well it spells out $\mathcal{L}_X$ in terms of contraction and the exterior derivative, it's very user-friendly. In particular you can establish easy results, for instance when $\omega$ is a closed symplectic form and $X$ is the Hamiltonian vector field.2012-08-30
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    Either one of the Cartans, [father Élie](http://en.wikipedia.org/wiki/Élie_Cartan) or [son Henri](http://en.wikipedia.org/wiki/Henri_Cartan) could plausibly have invented it... See also [the MO thread *Is “Cartan’s magic formula” due to Élie or Henri?*](http://mathoverflow.net/questions/39540/is-cartans-magic-formula-due-to-lie-or-henri)2012-08-30
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    Yes, I've seen the MO thread, in fact. The parenthetical question at the end of the post is not mine, but was added by others.2012-08-30
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    Maybe because $\mathscr{L}_X$ is of $0$-degree (derivation) and is nicely related to a $1$-degree map $d$ and a $-1$-degree map $i_X$2013-02-21

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I would have liked to put this as a comment but I do not have enough points to do it.

I think there is some magic in this formula because it tells us that the Lie derivative $\mathscr{L}_X$ is homotopic to zero with the homotopy $i_X$ going from top-right $\Omega^{p+1}(M)$ to bottom-left $\Omega^p(M)$ diagonally in the following diagram:

$$\require{AMScd} \begin{CD} \cdots @>{d}>> \Omega^p(M) @>{d}>> \Omega^{p+1}(M) @>{d}>> \cdots \\ \qquad @V{\mathscr{L}_X}V{0}V @V{\mathscr{L}_X}V{0}V\\ \cdots @>{d}>> \Omega^p(M) @>{d}>> \Omega^{p+1}(M) @>{d}>> \cdots \end{CD}$$