Help with an specific example of a higher derivative Lagrangian

I want to find the equation of motion that comes from the following Lagrangian density $$\mathscr{L}=\mathbf{E}\cdot\left(\nabla^{2}\mathbf{E}\right)$$ where $$E_{i}=\partial_{i}\phi\;(i=x,y,z)$$ . In this case the $$\phi$$ and its 3rd derivatives are the independent variables. The Euler-Lagrange equation contains two terms $$\partial_{i}\frac{\mathscr{\partial L}}{\partial(\partial_{i}\phi)}$$ and $$\partial_{ijk}\frac{\mathscr{\partial L}}{\partial(\partial_{ijk}\phi)}.$$

For the first term I'm getting $$\partial_{i}\frac{\mathscr{\partial L}}{\partial(\partial_{i}\phi)}=\nabla\cdot\left(\nabla^{2}\mathbf{E}\right),$$ a nice scalar function.

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Now, Im having problem with the 3rd derivatives term.

QUESTION: Are permutation of the index to be considered independent variables or the same?, e.g, it is $$\partial_{xxy}\phi$$ the same independent variable as $$\partial_{xyx}\phi$$?

If different permutations are to be understood as the same, I get the following $$\partial_{ijk}\frac{\mathscr{\partial L}}{\partial(\partial_{ijk}\phi)} = \;\partial_{xxx}E_{x}+3\partial_{xyy}E_{x}+3\partial_{xzz}E_{x} +3\partial_{xxy}E_{y}+\partial_{yyy}E_{y}+3\partial_{zzy}E_{y} +3\partial_{xxz}E_{z}+3\partial_{zyy}E_{z}+\partial_{zzz}E_{z},$$

which doesn't seems to be an scalar.

QUESTION: Is my result correct? If not, why not (obviously)? If yes, then the last expression should be expressed in an scalar combination of $$\nabla$$ that I'm not seeing it, correct?

EDIT: If,on the other hand, I should only have to consider one permutation of $$\partial_{xxy}\phi$$, then the result seems to be

$$\partial_{ijk}\frac{\mathscr{\partial L}}{\partial(\partial_{ijk}\phi)} = \;\partial_{xxx}E_{x}+\partial_{xyy}E_{x}+\partial_{xzz}E_{x} +\partial_{xxy}E_{y}+\partial_{yyy}E_{y}+\partial_{zzy}E_{y} +\partial_{xxz}E_{z}+\partial_{zyy}E_{z}+\partial_{zzz}E_{z}=\nabla\cdot\left(\nabla^{2}\mathbf{E}\right),$$

• ... you know you can integrate by parts, and reduce the problem to first order derivatives only, right? – AccidentalFourierTransform Apr 28 at 15:23