Single-Particle Electrodynamics - Assassination Science

n via the inverse of (6.5), namely,
n ≡ R R .
By using (6.51) and (6.49), one finds that
n = n d − 1 2 r d ˙v + 1 2 r d(n d· ˙v)n d + 1 6 r2 d ¨v − 1 8 r2 d ˙v2 n d + 1 8 r2 d (n d· ˙v) 2 n d
− 1 6 r2 d (n d·¨v)n d − 1 4 r dr s (n d· ˙v)(n s· ˙v)n d + 1 4 r dr s (n s· ˙v) ˙v + · · · + O(ε 5 ).
6.5.5 Retarded velocity and derivatives
Taking successive t-derivatives of (6.46), we have
v r ′(t) = t ˙v + 1 2 t2¨v − t(r ′· ˙v) ˙v + 1 6 t3 ...
v − t 2 (r ′· ˙v)¨v − 1 2 t2 (r ′·¨v) ˙v
+ t(r ′· ˙v) 2 ˙v + 1 24 t4 ....
v − 1 2 t3 (r ′· ˙v) ... v − 1 2 t3 (r ′·¨v)¨v − 1 6 t3 (r ′·... v) ˙v
− 1 2 t3 ˙v 2 (r ′· ˙v) ˙v + 3 2 t2 (r ′· ˙v) 2¨v + 3 2 t2 (r ′· ˙v)(r ′·¨v) ˙v − t(r ′· ˙v) 3 ˙v
(6.52)
+ O(ε 5 ), (6.53)
˙v r ′(t) = ˙v + t¨v − (r ′· ˙v) ˙v + 1 2 t2 ...
v − 2t(r ′· ˙v)¨v − t(r ′·¨v) ˙v + (r ′· ˙v) 2 ˙v
+ 1 6 t3 ....
v − 3 2 t2 (r ′· ˙v) ... v − 3 2 t2 (r ′·¨v)¨v − 1 2 t2 (r ′·... v) ˙v − 3 2 t2 ˙v 2 (r ′· ˙v) ˙v
+ 3t(r ′· ˙v) 2¨v + 3t(r ′· ˙v)(r ′·¨v) ˙v − (r ′· ˙v) 3 ˙v + O(ε 4 ), (6.54)
¨v r ′(t) = ¨v + t ... v − 2(r ′· ˙v)¨v − (r ′·¨v) ˙v + 1 2 t2 ....
v − 3t(r ′· ˙v) ... v − 3t(r ′·¨v)¨v
− t(r ′·... v) ˙v − 3t ˙v 2 (r ′· ˙v) ˙v + 3(r ′· ˙v) 2¨v + 3(r ′· ˙v)(r ′·¨v) ˙v
+ O(ε 3 ). (6.55)
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