Euler-Lagrange Equations for a field

Given the functional:

a. find the canonical mementa and construct the Hamiltonian tensor components, and

b. write the Euler-Lagrange equations

Notation:

Now I think can do part a (soln below) but I am strugling with part b. Obviously if I have part a wrong then that would make part b "difficult"

Part a.

The canonical momentum is

The Hamiltonian tensor is

Part b.

The functional does not depend explicitly on the coordinates or on phi. So the Euler-Lagrange equations are:

and

I have tried all kinds of things but they always become computationally complex and that seems unlikely.