Bill Hobba wrote:
Now L (interaction) may depend on position, time, 4 velocity and charge.
From the symmetry properties of an inertial reference frame is should not
depend on position or time thus should only depend on the 4 velocity. If
we
assume it is a linear theory then it depends linearly on charge and 4
velocity ie has the form (in units where c is one) (-qAidx/dt)dt (dx/dt is
the four velocity) where Ai is the constant of proportionality. Since q
and
the 4 velocity are associated with the charged particle Ai must be
associated wit with field and is called the four potential. Thus the
lagrangian of the particle and the field interaction is:

-mcds - q/cAidxi .

Taking the variation we find the equations of motion of a charged particle
to be dp/dt = eE + e/c VxH where the electric and magnetic field are
defined
in the usual way from the four potential. (see page 47 of Landau Classical
Theory of Fields for the full working out).


I goofed. I said we take the speed of light as one and used q as the
charge. The lagrangian should be of course:

-mds -qAidxi

with the variation giving the equations of motion as:

dp/dt = qE + qVxH

Sorry
Bill