Single-Particle Electrodynamics - Assassination Science

G.6 radreact: Radiation reaction
G.6.1
Introduction
This program computes the radiation reaction equations of motion for particles
carrying electric charge and electric and magnetic dipole moments.
G.6.2
Pointlike particle trajectory
The following four-vector expression is an input to the program, computed
by the author on paper:
t(τ) = τ + 1 6 τ 3 ˙v 2 + 1 8 τ 4 ( ˙v·¨v) + 13
120 τ 5 ˙v 4 + 1
40 τ 5¨v 2 + 1
30 τ 5 ( ˙v·... v)
+ 1
144 τ 6 ( ˙v·.... v) + 1
72 τ 6 (¨v·... v) + 3
16 τ 6 ˙v 2 ( ˙v·¨v) + O(τ 7 ),
z(τ) = 1 2 τ 2 ˙v + 1 6 τ 3¨v + 1
24 τ 4 ...
v + 1 6 τ 4 ˙v 2 ˙v + 1
120 τ 5 ....
v + 1 12 τ 5 ˙v 2¨v
+ 1 8 τ 5 ( ˙v·¨v) ˙v + O(τ 6 ), (G.14)
Although these expressions are tedious to derive, they are easy to verify.
Firstly, we compute ż 2 (τ), by differentiating (G.14) with respect to τ:
ż 2 (τ) = 1,
the required result. We now compute v(τ) ≡ d τ z(τ)/d τ t(τ):
v(τ) = τ ˙v + 1 2 τ 2¨v + 1 6 τ 3 ...
v + 1 6 τ 3 ˙v 2 ˙v + 1 24 τ 4 ....
v + 1 6 τ 4 ˙v 2¨v + 1 8 τ 4 ( ˙v·¨v) ˙v
+ O(τ 5 ); (G.15)
computing γ(τ) ≡ (1 − v 2 (τ)) −1/2 , we find
γ(τ) = 1 + 1 2 τ 2 ˙v 2 + 1 2 τ 3 ( ˙v·¨v) + 13
24 τ 4 ˙v 4 + 1 8 τ 4¨v 2 + 1 6 τ 4 ( ˙v·... v)
+ 1
24 τ 5 ( ˙v·.... v) + 1
12 τ 5 (¨v·... v) + 9 8 τ 5 ˙v 2 ( ˙v·¨v) + O(τ 6 ). (G.16)
446