GPS 8.12 (2
nd
ed.
82
)
Start with the kinetic energy written as the sum of the center of mass con
tribution and the contribution from motion relative to the COM:
T
=
M
2
·
R
2
+
m
1
2
μ
·
r
0
1
¶
2
+
m
2
2
μ
·
r
0
2
¶
2
,
(1)
where
M
=
m
1
+
m
2
,
m
i
is the mass of particle
i
,
R
is the COM position vector,
and
r
0
i
is the position vector of particle
i
relative to the COM. With the use of
the relative position vector
r
r
=
r
0
2
−
r
0
1
,
(2)
the expression for
T
simpli
f
es (cf. Chap. 3) to
T
=
M
2
·
R
2
+
μ
2
·
r
2
,
(3)
where
μ
is the reduced mass of the twobody system
μ
=
m
1
m
2
M
.
(4)
The two particles interact with a central force so the potential energy of the
system is simply
V
(
r
)
. Hence, the Lagrangian is
L
=
M
2
·
R
2
+
μ
2
·
r
2
−
V
(
r
)
.
(5)
Because the COM coordinates are cyclic variables, the components of the linear
COM momentum are conserved quantities, and as a result the COM kinetic
energy is just an additive constant in the Lagrangian that we will ignore from
now on.
In spherical coordinates the components of
r
are
x
=
r
sin
θ
cos
φ,
(6)
y
=
r
sin
θ
sin
(7)
z
=
r
cos
θ,
(8)
where
r
=

r

,
θ
is the polar angle between
r
and the
z
axis, and
φ
is the
azimuthal angle measured from the
x
axis in the
x

y
plane. After transforming
into spherical coordinates the Lagrangian reads
L
=
μ
2
[
·
r
2
+
r
2
·
θ
2
+
r
2
·
φ
2
sin
2
θ
]
−
V
(
r
)
.
(9)
Next, using the prescription,
p
j
=
∂L/∂
·
q
j
,we
f
nd the following results for the
three conjugate momenta
p
r
,
p
θ
,and
p
φ
:
p
r
=
μ
·
r,
(10)
1
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θ
=
μr
2
·
θ,
(11)
and
p
φ
=
μr
2
·
φ
sin
2
θ.
(12)
Next we construct the energy function
h
,de
f
ned here as
h
=
·
r
∂L
∂
·
r
+
·
θ
∂
·
θ
+
·
φ
∂
·
φ
−
L.
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 Fall '10
 Wilemski
 mechanics, Angular Momentum, Center Of Mass, Energy, Kinetic Energy, Mass, Sin, Coordinate system, Spherical coordinate system, Polar coordinate system

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