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Single-Particle Electrodynamics - Assassination Science

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Expanding out these expressions explicitly, we find<br />

E ′q<br />

1 = −κ 2 ˙v − κ 3 ( ˙v·n)v + κ 3 ( ˙v·n)n,<br />

E ′q<br />

2 = −κ 3 γ −2 v + κ 3 γ −2 n,<br />

E ′d<br />

1 = −κ 2 ¨σ ′ − κ 3 (σ ′·n)¨v − 2κ 3 ( ˙σ ′·n) ˙v − κ 3 (¨σ ′·n)v + κ 3 (¨σ ′·n)n<br />

− κ 3 ( ˙v·n) ˙σ ′ − 3κ 4 (σ ′·n)( ˙v·n) ˙v − κ 4 (σ ′·n)(¨v·n)v<br />

+ κ 4 (σ ′·n)(¨v·n)n − 3κ 4 ( ˙σ ′·n)( ˙v·n)v + 3κ 4 ( ˙σ ′·n)( ˙v·n)n<br />

− 3κ 5 (σ ′·n)( ˙v·n) 2 v + 3κ 5 (σ ′·n)( ˙v·n) 2 n,<br />

E ′d<br />

2 = −κ 3 γ −2 ˙σ ′ + 2κ 3 (σ ′·v) ˙v + κ 3 (σ ′· ˙v)v − κ 3 (σ ′· ˙v)n + κ 3 (σ ′·n) ˙v<br />

+ 2κ 3 ( ˙σ ′·v)v − 2κ 3 ( ˙σ ′·v)n − κ 3 ( ˙v·n)σ ′ − 3κ 4 γ −2 (σ ′·n) ˙v<br />

− 3κ 4 γ −2 ( ˙σ ′·n)v + 3κ 4 γ −2 ( ˙σ ′·n)n + 3κ 4 (σ ′·v)( ˙v·n)v<br />

− 3κ 4 (σ ′·v)( ˙v·n)n + 3κ 4 (σ ′·n)(v· ˙v)v − 3κ 4 (σ ′·n)(v· ˙v)n<br />

− 6κ 5 γ −2 (σ ′·n)( ˙v·n)v + 6κ 5 γ −2 (σ ′·n)( ˙v·n)n,<br />

E ′d<br />

3 = −κ 3 γ −2 σ ′ + 3κ 4 γ −2 (σ ′·v)v − 3κ 4 γ −2 (σ ′·v)n − 3κ 5 γ −4 (σ ′·n)v<br />

+ 3κ 5 γ −4 (σ ′·n)n,<br />

B ′q<br />

1 = κ 2 ˙v×n + κ 3 ( ˙v·n)v×n,<br />

B ′q<br />

2 = κ 3 γ −2 v×n,<br />

B ′d<br />

1 = κ 2 ¨σ ′ ×n + κ 3 (σ ′·n)¨v×n + 2κ 3 ( ˙σ ′·n) ˙v×n + κ 3 (¨σ ′·n)v×n<br />

+ κ 3 ( ˙v·n) ˙σ ′ ×n + 3κ 4 (σ ′·n)( ˙v·n) ˙v×n + κ 4 (σ ′·n)(¨v·n)v×n<br />

+ 3κ 4 ( ˙σ ′·n)( ˙v·n)v×n + 3κ 5 (σ ′·n)( ˙v·n) 2 v×n,<br />

444

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