6Systems of particles
IA Dynamics and Relativity
6.2 Motion relative to the center of mass
So far, we have shown that externally, a multiparticle 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.