Lecture 2: Two-body problem (4 Sep 09)
A. Relative motion of two bodies
1. Use notation
R
≡
~
R
[bold face for vectors] so that dot for time deriva-
tive is
˙
R
.
2. Formulate Newton’s laws using two bodies with internal forces – second
and third laws:
d
p
1
/dt
=
F
21
;
F
21
=
-
F
12
and central forces
F
21
= [
-
d
Φ
/dr
](
r
1
-
r
2
)
/r
12
= [
-
d
Φ
/dr
]ˆ
r
12
.
3. Center-of-mass and relative coordinates defined for masses
m
1
and
m
2
,
M
=
m
1
+
m
2
: (L & L, Sec. 13)
R
=
m
1
r
1
+
m
2
r
2
M
;
r
=
r
1
-
r
2
with inverse relations
r
1
=
R
+
m
2
M
r
;
r
2
=
R
-
m
1
M
r
4. The corresponding relations for center-of-mass momentum and relative
momentum are (anticipating quantum mechanics)
P
=
p
1
+
p
2
;
p
=
m
2
p
1
-
m
1
p
2
M
p
1
=
m
1
M
P
+
p
;
p
2
=
m
2
M
P
-
p
5. For the mechanical momenta
p
j
=
m
j
˙
r
j
, the relative momentum is,
with reduced mass
μ
:
p
=
μ
˙
r
;
μ
=
m
1
m
2
m
1
+
m
2
;
1
μ
=
1
m
1
+
1
m
2
6. Conservation of total (center-of-mass) momentum (only internal forces)
d
dt
P
=
F
21
+
F
12
= 0
1
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using Newton #3
⇒
explicit solution for
R
(
t
).
0 =
d
dt
P
=
M
d
2
dt
2
R
⇒
R
=
R
0
+
V
t
where
V
is the constant center-of-mass velocity.
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