Bookmark 3 :
J
Angular momentum with respect to the origin and spin
S
, angular momentum with respect to the CC (
r
)
The free invariant Lagrangian
L
o
is independent of t and
r
,​​
L
0
=
T
0
+
R
0
u
+
U
0
a
+
W
0
ω
​​and the functions
T
0
,
R
0
,
U
0
and
W
0
, are still unknown functions of (
u
,
a
,
ω
).

9. The Center of Mass is different from the Center of Charge


10. Expressions for the Spin of the particle

We will now derive the expression for the Spin of the particle.If we make the cross product with
u
of the expression (15) we get​​
1
2
c

S

u

u
=
1
2
c
H
k

u
Using
B

C

A
=(
A
·
B
)
C
-
A
·
C

B
and
A

B

C
=
A
·
C

B
-(
A
·
B
)
C
(
19
)
and
u
·
u
=
2
c
​​​

u
·
S

u
-
2
c
S
=H
r
-
q

u
As we shall see in (26) the velocity
u
and
S
are orthogonal vectors (
u
·
S
=0) and we obtain using (17) the explicit expression of the spin with respect to the CC
S
=-γ(
v
)m
r
-
q

u
(
20
)
Because we have another point
q
(CM), the spin with respect to the point CM is defined as the spin
S
in the point
r
plus the vector from
r
to
q
cross product with the Linear Momentum
P
:
S
CM
=
S
+
r
-
q

P
​​
S
CM
=-γ(
v
)m
r
-
q
(
u
-
v
)
(
21
)
Both spins
S
&
S
CM
satisfy different dynamical equations. ​The total angular momentum
J
of the particle with respect to the origin of any arbitrary inertial frame from (13), can be written using (21) in two different ways as:
J
=
r

P
+
S
​​
J
=
q

P
+
S
CM
(
22
)