Figure 1:

v
defined by its module v, β the zenithal angle and λ the azimuthal angle. Spin
S
CM
is defined by its zenithal angle θ and azimuthal angle ϕ. ψ is the phase of the CC.
We choose a laboratory observer that describes the two-particle system in their common center of mass frame, then the CM of the two particles is at rest and located at the origin of the reference frame.
Since the Lagrangian (13) is invariant under space translations, the total linear momentum is conserved and the motion of the particles is such that, in the Coulomb interaction,
q
1
(t) = −
q
2
(t) and
v
1
(t) = −
v
2
(t) .
In the Poincaré invariant interaction
γ(
v
1
)
q
1
(t)+
γ(
v
2
)
q
2
(t) = 0 holds but
p
T
= 0 does not imply that
v
1
(t) = −
v
2
(t). This requirement will be explicitly determined in the analysis of this interaction in section 8.
If
*
t
and
*

r
are the time and position of the CC of one of the particles for the center of mass observer
*
O
of this particle, and t and
r
are the time and position of the CC for any arbitrary inertial observer, they are related by the Poincaré transformation: (Part I.Section4)
μ
x
=
µ
Λ
ν
*ν
x
+
μ
a
μ
x
≡(ct,

r
)
*μ
x
≡(
*
ct
,
*

r
)
μ
a
≡cb,
d

and Λ = L(
v
)R(ψ, θ, ϕ) is a general Lorentz transformation, as a composition of a rotation R, followed by a boost of velocity
v
, or pure Lorentz transformation L(
v
).
We follow now the same derivation as in the previous post (Part I.Section17 Dynamic equations and Particle Initial Conditions) to obtain:
r
0
=
*
r
0
-
γ
(1+γ)

v
*
r
0

v
-b
v
+
d
​​
u
0
=
*
u
0
+γ
v
+
2
γ
(1+γ)

v
·
*
u
0

v
γ1+
v
*
u
0

​​
q
0
=
v
*
r
0

*
u
0
-
v
*
u
0

*
r
0
-b
v
+
d
=
v
×
*
u
0
×
*
r
0
−b
v
+
d
​​
v
0
=
v
(
30
)
with
*
r
0
=
1
2
Cos[ϕ]Cos[θ]Cos[ψ]-Sin[ϕ]Sin[ψ]
Sin[ϕ]Cos[θ]Cos[ψ]+Cos[ϕ]Sin[ψ]
-Sin[θ]Cos[ψ]
(
31
)
*
u
0
=
Cos[ϕ]Cos[θ]Sin[ψ]+Sin[ϕ]Cos[ψ]
Sin[ϕ]Cos[θ]Sin[ψ]-Cos[ϕ]Cos[ψ]
-Sin[θ]Sin[ψ]
(
32
)
*
a
0
=-2
Cos[ϕ]Cos[θ]Cos[ψ]-Sin[ϕ]Sin[ψ]
Sin[ϕ]Cos[θ]Cos[ψ]+Cos[ϕ]Sin[ψ]
-Sin[θ]Cos[ψ]
=-4
*
r
0
(
33
)
a
0
=
1
2
γ
3
1+
v
*
u
0

1+
v
*
u
0

*
a
0
-
v
*
a
0

*
u
0
-
γ
(1+γ)

v
*
a
0

v
(
34
)