Independent variables in higher derivative Lagrangian

[andresB]

Consider 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 th
is case the potential and its 3rd derivatives are the independent variables. Acording to Barut's classical theory of fields book, for the above system the Euler-Lagrange equation contains two terms
$$\partial_{i}\frac{\mathscr{\partial L}}{\partial(\partial_{i}\phi)}$$ and $$\partial_{ijk}\frac{\mathscr{\partial L}}{\partial(\partial_{ijk}\phi)}.$$The issue is the following: are different permutation of the indices to be considered the same variable or different variables?, for example, Asuming a well-behaved function we have that $$\partial_{xxy}\phi=\partial_{xyx}\phi=\partial_{yxx}\phi,$$
but should the term $$\partial_{xxy}\frac{\mathscr{\partial L}}{\partial(\partial_{xxy}\phi)}$$ appear once or three times in the equation of motion?

 

[eljose79]

it is important to consider the underlying principles and assumptions of the theory before making any conclusions. In this case, the Lagrangian density is a mathematical expression that describes the dynamics of a physical system. It is constructed from the potential and its derivatives, which are considered as independent variables.

In the context of this Lagrangian density, the potential and its third derivatives are indeed considered as independent variables. This means that different permutations of the indices should be treated as different variables. Therefore, the term $$\partial_{xxy}\frac{\mathscr{\partial L}}{\partial(\partial_{xxy}\phi)}$$ should appear three times in the equation of motion.

This may seem counterintuitive, but it is important to remember that the Lagrangian density is a mathematical construct and not a physical quantity. It is a tool used to describe the behavior of a physical system and is based on certain assumptions and principles. In this case, the assumption is that the potential and its third derivatives are independent variables, and this should be reflected in the equation of motion.

In conclusion, different permutations of the indices should be considered as different variables in the equation of motion for this Lagrangian density. It is important to carefully consider the underlying assumptions and principles of a theory before making any conclusions or interpretations.