Into Electrodynamics

With the notational ideas cleared up, i move into the covariant formulation of electrodynamics.But first, a short review of relativistic mechanics.
Using the principle of least action,the lagrangian is determined to be
L=-mc^2*sqrt(1-v^2/c^2)
Then we go ahead to find the expressions for momentum and energy in the case of a free particle.the euler-lagrange equations of motion(ELEOM) give us the condition that the four-velocity must be a constant.
LL makes the comment that the energy in relativistic mechanics is always positive and proportional to the mass, unlike in class mech where it can be either positive or negative.That would be because in class mech u do have the potential energy of interaction of a system of particles.but to do so in rel. mech. one has to explicitly include the momentum and energy of the field produced by the interacting particles.
A short review was also made of the angular momentum tensor, its antisymmetric nature.

Next, we come to the section where the field is viewed as an physical entity that mediates interactions between particles, since relativity allows only contact interaction.A particle imparts the information about its motion to the field surrounding it, which then propagate it far away.Also, since relativiy disallows the concept of rigid bodies, the so called elementary particles must be pointlike.
the next sections are more mathematical.in them we use the four-vector A to characterise the field, obtain the lagrangian, the generalised momentum,the hamiltonian.Then we proceed further to obtain the ELEOM and define the elecric and magnetic fields.interestingly, we see that the work done on the particle is only by the electric field E, since the magnetic field B is always perpendicular to the velocity v.
Equations of motion in classical mechanics are time reversal invariant; the same thing is true in electrodynamics under the transformation t-> -t; E -> E ; B -> -B which in turn implies phi -> phi; A -> -A; phi and A being the components of the four vector A.
then i studied gauge invariance and solutions for E,B and energy in constant electomagnetic fields.