Say you have a lagrangian $F$ for a system:
$J[y,z] = \int_a^bF(x,y,y',z,z')dx \tag{1}$
If y and z are associated with two parts of the system and the parts are distant enough that the interaction between them can be neglected, then the Euler-Lagrange equation wrt $y$ and $y'$ must not contain $z$ and $z'$, and vice versa. I'd then like to prove that $F$ can be written as:
$F(x,y,y',z,z') = F_1(x,y,y') + F_2(x,z,z') \tag{2}$
Here, $F_y$ and $F_{y'}$ are not functions of $z$ and $z'$. Similarly, $F_z$ and $F_{z'}$ are not functions of $y$ and $y'$. I intuitivley know that this must mean that y and z terms in $F$ must be separate (no coupling). But I'd like to rigorously prove this. Can I use $F_{yz}=0$ and so on, in doing so?