6Systems of particles

IA Dynamics and Relativity

6.2 Motion relative to the center of mass
So far, we have shown that externally, a multi-particle system behaves as if it
were a point particle at the center of mass. But internally, what happens to the
individual particles themselves?
We write
r
i
=
R
+
r
c
i
, where
r
c
i
is the position of particle
i
relative to the
center of mass.
We first obtain two useful equalities:
X
i
m
i
r
c
i
=
X
m
i
r
i
X
m
i
R = MR MR = 0.
Differentiating gives
X
i
m
i
˙
r
c
i
= 0.
Using these equalities, we can express the angular momentum and kinetic energy
in terms of R and r
c
i
only:
L =
X
i
m
i
(R + r
c
i
) × (
˙
R +
˙
r
c
i
)
=
X
i
m
i
R ×
˙
R + R ×
X
i
m
i
˙
r
c
i
+
X
i
m
i
r
c
i
×
˙
R +
X
i
m
i
r
c
i
×
˙
r
c
i
= MR ×
˙
R +
X
i
m
i
r
c
i
×
˙
r
c
i
T =
1
2
X
i
m
i
|
˙
r
i
|
2
=
1
2
X
i
m
I
(
˙
R +
˙
r
i
c
) · (
˙
R +
˙
r
c
i
)
=
1
2
X
i
m
i
˙
R ·
˙
R +
˙
R ·
X
i
m
i
˙
r
c
i
+
1
2
X
i
m
i
˙
r
c
i
·
˙
r
c
i
=
1
2
M|
˙
R|
2
+
1
2
X
i
m
i
|
˙
r
c
i
|
2
We see that each item is composed of two parts that of the center of mass
and that of motion relative to center of mass.
If the forces are conservative in the sense that
F
ext
i
= −∇
i
V
i
(r
i
),
and
F
ij
= −∇
i
V
ij
(r
i
r
j
),
where
i
is the gradient with respect to
r
i
, then energy is conserved in the from
E = T +
X
i
V
i
(r
i
) +
1
2
X
i
X
j
V
ij
(r
i
r
j
) = const.