Single-Particle Electrodynamics - Assassination Science

4.3.4 Relativistic Lagrangian derivation
We shall now briefly review another method that the author [65] has used
to derive the manifestly covariant equations of motion for point particles
carrying (fixed) electric charge and electric and magnetic dipole moments.
(The full text of this paper is given in Appendix F, but it should be noted
that the redshift force does not appear therein, and hence the “uncoupled”
equations of motion listed at the end are incorrect.)
In this method of derivation, one essentially recognises from the outset
that one wishes the magnitudes of the dipole moments to be fixed.
One
therefore writes down a relativistic Lagrangian in such a way that the dipole
moments are treated like four-vectors of fixed magnitude. One way to do so,
recognising the fundamental practical importance of spin-half particles, is to
start from the most general interaction Lagrangian possible between such
particles and the electromagnetic field, from the point of view of quantum
field theory; and then to massage the functional form of the resulting Lagrangian
somewhat. This procedure does not lead to any new results, and is
somewhat tangential to the main thrust of this thesis, and so has been relegated
to Appendix E. The form of the relativistic Lagrangian found there
is
L = 1 2 m(U 2 ) + q(U ·A) + (d·F ·U) + (µ· ˜F ·U).
The Euler–Lagrange equations for the four translational degrees of freedom
z α give
d τ (mU) = −qd τ A − d τ (d·F ) − d τ (µ· ˜F )
+ q∂(U ·A) + ∂(d·F ·U) + ∂(µ· ˜F ·U). (4.70)
Using the identities (B.27), (B.28) and (B.29), one then immediately obtains
the result
ṗ = qF · U + (d·∂)F · U + [F · ˙ d ] + (µ·∂) ˜F · U + [ ˜F · ˙µ ] + µ×J ×U, (4.71)
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