Single-Particle Electrodynamics - Assassination Science

i.e., the torque. Clearly, the corresponding Euler–Lagrange equations will be
those due to the Euler angle generalised coördinates of the rest-frame threespin
s ≡ sσ, as in the nonrelativistic case [96]. But relativistic generalisations
of these Euler angles do not appear to be worth the complications involved,
since they do not even form a three-vector in the rest frame [96], let alone
a four-vector in the arbitrarily moving frame; thus, any manifestly covariant
derivation must arguably employ the procedure of constraints [96] on the
relativistic generalisation Ω α of the nonrelativistic angular frequency vector
ω, which contains combinations of the time-derivatives of the Euler angles.
The author has also (despite attempts apparently successful, but then found
to be wanting) been unable to formulate any relativistic version of the Euler–
Lagrange equations for the explicit Euler-angle expressions, in any fashion
that might reasonably be described as “transparent”.
Fortunately, the torque equation of motion found in Chapter 4 will be
simply the Thomas–Bargmann–Michel–Telegdi equation [214, 25], which can
be derived on quite general physical grounds [108], and indeed has been
derived in so many different ways in the literature that it took the author
several days just to read them all. Since no modification to this famous and
well-loved law will be necessary, we shall be content, for the purposes of this
thesis, to simply follow the “relativistic bootstrap” procedure (described in
the next section) to obtain the relativistic torque equation of motion from
the nonrelativistic result; it is trusted that the reader will be satisfied with
this result, even if it might be considered preferable if a manifestly covariant
Lagrangian derivation, along the lines described in this section, could be
found. We leave the latter as an exercise for the reader!
2.6.12 The “relativistic bootstrap” process
One frequently wishes to obtain equations of motion for a physical system
that are relativistically correct, yet without needing to consider all of the
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