Lorentz force for a relativistic charge.

The Lorentz force for a charged particle is

Ok... now is where the magic of the physics begins. That equation seems like a horrible monster but I'm going to try to show a beauty face of it.

Think in a particle (e.g electron) moving forward in a region where you have electric or magnetic field, the first and natural question is how do these fields change the direction of the particle? find out.

Letting

these fields are constants and uniforms with that said we can start on it.

From the Lorentz force, the 0-component is

the i-component is

When we've done with all the maths, will have this pair of equations

For the problem of interest the pair of equations will reduce to

So the final equations for the component 2 of the 4-velocity is

Setting the initial conditions

The world line will be,

I've omitted the 3 component because it's a constant(straight line).

At the end the velocity is of course changed because of the electromagnetic field.

The next and last step is just plot how is the movement of a charged particle due to a electromagnetic field.

The hyperbolic motion looks

The dominance of the electric field kills the harmonic motion.