Single-Particle Electrodynamics - Assassination Science

can be written in manifestly covariant terms.
The former question will be answered in more detail in Section 2.6.12;
basically, as long as one can obtain a valid Lagrangian description that is
dynamically correct to first order in the velocity of the particle, then there is
a definite procedure that one may follow to “bootstrap” the resulting equations
of motion, unambiguously, into the relativistic domain; the kinematical
consequences of Einsteinian mechanics are then automatically included correctly.
If one wishes to actually write down a Lagrangian that itself automatically
yields the correct equations of motion to all orders in the particle’s velocity,
but by still using the lab-time t of the Euler–Lagrange equations (2.22), then
one is generally advised to begin with the “bootstrap” procedure above, and
thence seek a Lagrangian function that reproduces the equations of motion.
It is indeed possible to find such functions for simple situations; e.g., for a
free particle, one may use [96, Sec. 7–8]
L = −m √ 1 − v 2 ;
for an electric charge, one notes that the Lorentz force law is actually correct
relativistically, and hence the electromagnetic part of the Lagrangian (2.42)
still suffices. However, it is not clear [96] whether such a procedure can be
made to work for arbitrary systems of relativistic particles.
The second question above is whether one can find a relativistic Lagrangian
framework that is manifestly covariant. The subtlety arising here
(see, e.g., [96, Sec. 7–9]) is that one should seek to use the proper time τ
as the “time” parameter in the variational framework. For constant mass
particles, one then faces the problem that the four translational degrees of
freedom z α are actually constrained by the identity (2.53):
(d τ z) 2 = 1; (2.75)
hence, there are effectively only three translational degrees of freedom for
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