Single-Particle Electrodynamics - Assassination Science

3.3.5 Lab-time constituent trajectories
We now obtain the trajectory of the constituent r as z r (t r ), without reference
to the body τ at all, by eliminating τ between equations (3.14) and (3.13).
Reverting (3.13), we have [4, eq. 3.6.25]
τ(t r ) = 1 λ t r − 1
2λ 3 (r·¨v)t2 r
+ 1 {
−λ(r·... v) − λ [ 1 + 4(r· ˙v) ] ˙v 2 + 3(r·¨v)
}t 2 3
6λ 5 r
+ 1 {
−λ 2 (r·.... v) − λ 2[ 3 + 13(r· ˙v) ] ( ˙v·¨v) − 15(r·¨v) 3
24λ 7
− 2λ [ 1 − 14(r· ˙v) ] ˙v 2 (r·¨v) + 10λ(r·¨v)(r·... v)
}
t 4 r
+ O(t 5 r). (3.22)
Using (3.22) in (3.14), we thus find that
z r (t) = r + 1
2λ ˙vt2 + 1 {
}
λ¨v − (r·¨v) ˙v
6λ 3
+ 1 {
λ 2 ...
v − λ(r·... v) ˙v − 3λ ˙v 2 (r· ˙v) ˙v − 3λ(r·¨v)¨v + 3(r·¨v) 2 ˙v
24λ 5
+ 1 {
λ 3 ....
v − λ 2 (r·.... v) ˙v − 6λ 2 (r·¨v) v ... − 4λ 2 (r·... v)¨v
120λ 7
t 3
− 12λ 2 ˙v 2 (r· ˙v)¨v − 8λ 2 ˙v 2 (r·¨v) ˙v − 10λ 2 (r· ˙v)( ˙v·¨v) ˙v
}
t 4
+ 15λ(r·¨v) 2¨v + 10λ(r·¨v)(r·... v) ˙v
}
+ 30λ ˙v 2 (r· ˙v)(r·¨v) ˙v − 15(r·¨v) 3 ˙v t 5 + O(t 6 ), (3.23)
where it is understood that t means t r . Equation (3.23) gives the trajectory
of the constituent r as an individual particle, without reference to the system
proper-time τ.
3.3.6 Spin degrees of freedom
We finally consider the spin degrees of freedom of the particle. The various
constituents r of the body each have a unit spin vector of their own, σ r ,
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