Single-Particle Electrodynamics - Assassination Science

2.6.10 The FitzGerald three-spin
In Section 2.6.7, we introduced the unit three-spin σ, that will be used
to represent any internal degrees of freedom for our classical particle that
may be fully described in terms a three-direction in its rest frame. We now
introduce another three-vector quantity, closely related to the three-spin σ,
but with subtle differences that will be found to simplify considerably a
number of explicit algebraic expressions that we will encounter in this thesis:
the FitzGerald three-spin,
σ ′ ≡ σ −
γ (v·σ)v. (2.74)
γ + 1
The reason for the author naming it after FitzGerald may be appreciated if
we compute its three-magnitude:
σ ′ 2 = 1 − (v·σ) 2 .
The magnitude of σ ′ is (like that of σ) unity, if σ lies in a plane perpendicular
to the three-velocity v; but it is contracted by a factor of
√
1 − v2 ≡ 1 γ
if σ lies parallel or antiparallel to the direction of v; in other words, σ ′ acts
like a FitzGerald–Lorentz contracted [87] version of σ.
The utility (indeed, the indispensability) of σ ′ as an algebraic tool will be
manifestly clear when we consider the retarded field expressions of Chapter 5.
(The author in fact employed the quantity σ ′ in the published paper
of Appendix F [65] as one of a number of “convenient quantities”, but did
not realise at the time that its simplifying properties are of quite a general
nature.)
2.6.11 Relativistic Lagrangian mechanics
There arise the dual questions of whether Lagrangian mechanics can be formulated
in a relativistically meaningful way, and whether such a formulation
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