There are not many ways to make a **scalar Lagrangian from the field tensor**.
We already know that

and we need to make our Lagrangian out of the fields, not just the current. Again,

One might consider

but that is a pseudo-scalar, not a scalar. That is, it changes sign under a parity transformation. The EM interaction is known to conserve parity so this is not a real option. As with the scalar field, we need to

The **Lagrangian for Classical Electricity and Magnetism** we will try is.

The next step is to check what the Euler-Lagrange equation gives us.

Note that, since we have four independent components of as independent fields, we have four equations; or one 4-vector equation. The

It is important to emphasize that we have a Lagrangian based, formal classical field theory for electricity and magnetism which
has **the four components of the 4-vector potential as the independent fields.**We could not treat each component of
as independent since they are clearly correlated.
We could have tried using the six independent components of the antisymmetric tensor but it would not have
given the right answer.
Using the 4-vector potentials as the fields does give the right answer.
**Electricity and Magnetism is a theory of a 4-vector field .**

We can also calculate the **free field Hamiltonian density**, that is, the Hamiltonian density in regions with no
source term.
We use the standard definition of the Hamiltonian in terms of the Lagrangian.

We just calculated above that

which we can use to get

We will study the interaction between electrons and the electromagnetic field with the Dirac equation.
Until then, the Hamiltonian used for non-relativistic quantum mechanics will be sufficient.
We have derived the Lorentz force law from that Hamiltonian.

Jim Branson 2013-04-22