Euler equation in a Riemannian manifold

I am trying to prove that the Euler equation for a Riemannian 3-manifold $$M$$ $$\frac{\partial u}{\partial t} + \nabla_{u} u = - \operatorname{grad} p \qquad \text (1)$$ is equivalent to $$\frac{\partial u^\flat}{\partial t} + \mathcal L_u u^\flat = \frac{1}{2} d(u^\flat(u)) - dp \qquad \text (2)$$ where $$u$$ is a (velocity) vector field, $$p$$ is a function on $$M$$ (pressure term), $$\mathcal L_u$$ is the Lie derivative along $$u$$, and $$\flat$$ is the musical isomorphism from the tangent bundle to the cotangent bundle.

By applying $$\flat$$ to (1) and recalling that $$\operatorname{grad} p = (dp)^\sharp$$ one easily arrives to $$\frac{\partial u^\flat}{\partial t} + (\nabla_{u} u)^\flat = - dp \qquad \text (3)$$ which is close already! Now the only step remaining is to show that $$$$(\nabla_{u} u)^\flat = \mathcal L_u u^\flat - \frac{1}{2} d(u^\flat(u)) \qquad (4)$$$$

I have arrived to the following local expression for the LHS in terms of the metric $$(g_{ij})$$, a local frame $$(\partial_i)$$ and its coframe $$(dx^i)$$, thus considering $$u=u^i \partial_i$$: $$(\nabla_{u} u)^\flat = g_{lm} \left( u^j\partial_j(u^m) + u^i u^j \Gamma^m_{ij} \right) dx^l \qquad \text (5)$$

My doubt is how can I arrive to the same expression for the RHS of (4). Thanks a lot for your help!

• Do you have a coordinate formula for the Lie derivative of a 1-form? Should be fairly straightforward from there. May 18, 2020 at 5:33
• @AnthonyCarapetis I don't. And I don't consider anything else on u: $u=u^i\partial_i$ May 18, 2020 at 7:58
• Well, what's your definition of $\mathcal L_u \theta$ when $\theta$ is a one-form? I could answer this question but it's only of use to you if it starts somewhere you're familiar with. May 18, 2020 at 8:31
• Sorry @AnthonyCarapetis I was a bit slept before and I didn't understand. I am working with the Cartan's formula: $\mathcal L_{X}\omega = \iota_X d \omega + d \iota_X \omega$, where $\iota_X(\omega) = \omega(X,\cdot)$ is the interior multiplication. With this, the RHS of (4) reads as $\iota_u d u^\flat + \frac{1}{2} d(u^\flat(u))$. I tried to express it as in (5) but I did not quite reached it, even when considering that $\Gamma_{cab} = \frac{1}{2}\, \left(\partial_{b}g_{ca} + \partial_{a}g_{cb} - \partial_{c}g_{ab}\right)$. May 18, 2020 at 9:21