Reviewing Basic Special Relativity

So, it didn't take very long to review the math needed to understand Maxwell's equations. I think I have a good grasp of the basic (though, as with mechanics, I will need more practice).

So on to special relativity. It has been a while since I took my modern physics course, but luckily Wikipedia, once again, has what I need for a basic review. 

Of course, let's start with the all-important Lorentz Factor:

Then the simple Lorentz Transformation for an inertial reference frame moving in the x direction...

Which yeilds time-dialation, and length-contraction in the direction of motion...

Which of-course has the inverse...

We also get the new way to add velocities...

And, we need definitions of Energy and momentum that'll be conserved...

We also need an reference frame invariant way to define "mass"...

If we try to relate the velocity four-vector  to the momentun four-vector  to get the analogous equation to the non-relativistic  we have...

Which leads to the famous equations...

Giving back our definitions of Energy and momentum. Cool huh?

But that was just a small digression from the straight summary of the basics (i.e. Special Relativity without the formalism)

So what do we have left? Force of course.  Which has quite a large number of forms.

We start with the definition that keeps...

...and just crank out from there...

Since , we have...

We can split that into components of acceleration, thus...

Well, that should cover the basics...well, there is Kinetic Energy. Real quick...

Let's look at the integral:

 

This allows us to conclude:

Well, that last bit was ugly, but the result was simple enough.

All this is a bit much for me to digest. But I think I got most of this (excepting maybe exactly knowing when to use inveriant mass, relativistic mass, or rest mass--but I'll figure it out).

After, I digest this stuff, it may be time to move to the formalism.

Re-Learning the Theories of Electricity and Magnatism

I am now in the process of changing from Engineer to Physicist. So, and reviewing a lot of my physics. It'll be a long road, and I will being going back to undergrad to do it. 

The laws of mechanics are straight forward enough.

Newton's second law:

Which is equivalent to the basic law in Lagrangian Mechanics:

Which is in-turn equivalent to the basic law's in Hamiltonian Mechanics:

Certainly, there are a lot of special cases to consider, the work energy theorem, the notion of conservative forces, etc. --I need plenty of practice, in addition.

But I will now focus on Electricity and magnetism, since it is used as a model to thing about many of the big problems in physics.  

The laws themselves are plenty....

There are the four equations for free charges:

Then the two new equations for total charge:

Beyond that each one has an integral form:

Beyond that, we need to comrehend that:

There is all that funkiness with the units of current density,  magnetizing field, electric displacement field, electric field, and magnetic field.  The accounting for that is something I don't have straight yet--not even conceptually.  

Mathematically, I need to review the meaning of the div and curl, as well as integrating over lines and surfaces. 

Beyond that, there is the task of learning how special relativity plays into things, and learning the covariant formlation of clasical electromagnatism.  But I suppose the last part can come while I try to tackle special relativity in full-form. 

So much to learn and so little time.