How to calculate the Lie derivative for 2-forms?

Question

I know how to calculate Lie derivative for one-forms I use this formula:

$$\mathcal{L}_X\alpha = \left(X^j\frac{\partial \alpha_i}{\partial \phi^j} + \alpha_j \frac{\partial X^j}{\partial \phi^i}\right)d\phi^i.$$

But how to calculate Lie derivative for 2-forms or 3-forms ? I think we can write for example a 2-form like a wedge product of 1-forms and then use formula: $$\mathcal L_X\omega_{1}\wedge\omega_{2}=(\mathcal L_X\omega_{1})\wedge\omega_{2}+\omega_{1}\wedge(\mathcal L_X\omega_{2})$$ But I tried and failed. Can someone explain me how to calculate Lie derivative for 2-forms and 3-forms ?

I wish to have an explicit formula or an explicit algorithm please.

Answer 1

You can use Cartan's homotopy formula:

$$\mathcal{L}_X\omega=d(\iota_X\omega)+\iota_Xd\omega,$$ where $d$ denotes the exterior derivative, $\iota_X$ denotes the interior product, and $X$ is a vector field.

EDIT- As for the interior multiplication:

Explicitly, if $X$ is a smooth vector field on a manifold $M$ and $\omega\in\Omega^k(M),$ then $\iota_X \omega$ is a $k-1$ form defined by $$(\iota_X\omega)(X_2,\cdots, X_k)=\omega(X,X_2,\cdots, X_k),$$ for any smooth vector fields $X_2,\cdots, X_k$.