References: Amitabha Lahiri & P. B. Pal, *A First Book of Quantum Field Theory*, Second Edition (Alpha Science International, 2004) – Chapter 2, Problem 2.1.

The Euler-Lagrange equations for a classical field are

where is the Lagrangian density, is the th field, are the generalized coordinates and the notation . As an example of how these equations can give rise to physical equations that are more familiar, we’ll look at the Lagrangian (I’ll leave off the ‘density’ to save space, but when talking about fields instead of particles, we’ll always mean Lagrangian *density*) for the electromagnetic field. At this point, we won’t worry about where this Lagrangian comes from; we’ll just state it as

where the electromagnetic field tensor is (This definition is taken from Moore’s book on general relativity and is actually the negative of Lahiri & Pal’s definition in their equation 8.6. We don’t actually use this form of the tensor in what follows so it doesn’t affect the result, but it’s important to realize that different authors use different definitions.)

the four-current is

and is the four-potential

In applying 1, we take the fields to be the potentials . What do the Euler-Lagrange equations give us for these fields?

The first term in 1 is just

The second term is a bit more involved. First, we need to write in terms of the four-potential, which is

In calculating the derivatives, we need to be careful to keep track of the positions (up or down) of the various indices. The in 1 are represented by with raised indices. The derivative becomes

That is, the index on the derivative is lowered and the index on the field is raised. To take the derivatives of we need to express in this form, so we get

where using Lahiri and Pal’s convention. Now we can calculate the inner derivative in the second term of 1. First, we’ll consider the field . Using the product rule we get (I’ll use for the coordinate index to avoid confusion with which is used to denote the component of ; remember also that is diagonal, and there is no implied sum on ):

When , the RHS is zero, and for , so

Plugging this into 1 we get, together with 6

This is the potential equivalent of the Maxwell equation (the system of units used in Lahiri & Pal takes ).

Now look at the field . We have

We get

Plugging this into 1 we have

Combining this with 6 gives

Doing the same calculation for and yields the and components, so we get

which is the other inhomogeneous Maxwell equation in potential form.