Single-Particle Electrodynamics - Assassination Science

+ O(ε 5 ), (G.43)
˙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 ), (G.44)
¨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 ). (G.45)
Substituting t = t ret from (G.32), by using (G.38) in (G.43), (G.44) and
(G.45), we find
v ret = −r d ˙v + 1 2 r2 d ¨v + 1 2 r dr s (n s· ˙v) ˙v − 1 ...
6 r3 d v − 1 8 r3 d ˙v2 ˙v − 1 8 r3 d (n d· ˙v) 2 ˙v
+ 1
12 r3 d (n d·¨v) ˙v − 1 2 r2 d r s (n s· ˙v)¨v − 1 4 r2 d r s (n s·¨v) ˙v − 1 4 r drs 2 (n s· ˙v) 2 ˙v
+ 1 ....
24 r4 d v + 1 8 r4 d ˙v2¨v + 1 8 r4 d (n d· ˙v) 2¨v − 1
12 r4 d (n d·¨v)¨v
− 1
d·...
24 r4 d (n v) ˙v + 1 12 r4 d ( ˙v·¨v) ˙v − 1 8 r4 d ˙v2 (n d· ˙v) ˙v
+ 1 8 r4 d (n d· ˙v)(n d·¨v) ˙v + 1 4 r3 d r s (n s· ˙v) ... v + 1 4 r3 d r s (n s·¨v)¨v
+ 1
12 r3 d r s (n s·... v) ˙v + 7 16 r3 d r s ˙v 2 (n s· ˙v) ˙v − 1
24 r3 d r s (n d· ˙v)(n s·¨v) ˙v
+ 3
16 r3 d r s (n d· ˙v) 2 (n s· ˙v) ˙v − 1 8 r3 d r s (n d·¨v)(n s· ˙v) ˙v + 3 8 r2 d r 2 s (n s· ˙v) 2¨v
+ 3 8 r2 d r 2 s (n s· ˙v)(n s·¨v) ˙v + 1 8 r dr 3 s (n s· ˙v) 3 ˙v + O(ε 5 ), (G.46)
˙v ret = ˙v − r d¨v + 1 2 r d(n d· ˙v) ˙v − 1 2 r s(n s· ˙v) ˙v + 1 ...
2 r2 d v − 1 2 r2 d (n d· ˙v)¨v
457