6Systems of particles
IA Dynamics and Relativity
6 Systems of particles
Now suppose we have
N
interacting particles. We adopt the following notation:
particle
i
has mass
m
i
, position
r
i
, and momentum
p
i
=
m
i
˙
r
i
. Note that the
subscript denotes which particle it is referring to, not vector components.
Newton’s Second Law for particle i is
m
i
¨
r
i
=
˙
p
i
= F
i
,
where F
i
is the total force acting on particle i. We can write F
i
as
F
i
= F
ext
i
+
N
X
j=1
F
ij
,
where
F
ij
is the force on particle
i
by particle
j
, and
F
ext
i
is the external force
on i, which comes from particles outside the system.
Since a particle cannot exert a force on itself, we have
F
ii
=
0
. Also, Newton’s
third law requires that
F
ij
= −F
ji
.
For example, if the particles interact only via gravity, then we have
F
ij
= −
Gm
i
m
j
(r
i
− r
j
)
r
i
− r
j

3
= −F
ji
.