Single-Particle Electrodynamics - Assassination Science

vanishes, for v = 0:
P | v=0
= 0; (2.83)
the postulates of relativity tell us that the power input is, in fact, the zero
component ṗ 0 of ṗ α . Equations (2.82) and (2.83), together with the postulate
that E is the electric part of the six-vector F αβ , and the definition of the fourvelocity
(G.3), finally do uniquely give us the relativistic expression for the
Lorentz force law:
ṗ = qF ·U.
The author may seem to be belabouring the point of ensuring all components
are computed fully; in the case of the Lorentz force, the (often overlooked)
fourth equation required is trivial; but, in general (and the dipoles will be
examples of this), this consideration is of the utmost importance.
Finally, we must point out one further complication to this “bootstrap”
process, when one wishes to use Lagrangian mechanics to obtain the nonrelativistic
result. The problem is that, in the course of using the Euler–Lagrange
equations, one takes the derivative of the Lagrangian with respect
to the (generalised) velocities. This operation effectively decreases the order
of v in all terms by one power. Since one usually wishes to perform
the “bootstrap” operation on the equations of motion themselves—not the
Lagrangian,—one must therefore retain an order of v in the initial nonrelativistic
Lagrangian treatment, in order that the resulting equations of motion
are correct to zeroth order.
On the surface of it, such a procedure would seem to run contrary to
our above insistence that Galilean kinematics is not, in general, correct to
even first order in the velocities. The author submits that this dilemma may
arguably be evaded on the grounds of a philosophical technicality: The generalised
velocities appearing in Lagrangian treatments are in fact required
for the purposes of dynamical considerations; the failure of the Galilean
framework noted above is in fact a kinematical deficiency. It is therefore
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