(a)
This is a problem of motion in a central force
f
eld determined by the
potential
V
(
r
)=
k
ln(
r/a
)
,
(1)
where
k
and
a
are constants. The Lagrangian is
L
=
m
2
μ
·
r
2
+(
r
·
θ
)
2
¶
−
k
ln(
r/a
)
.
(2)
Since
θ
is a cyclic variable,
∂L/∂
·
θ
is a constant of the motion, the angular
momentum
l
, and we can write immediately
l
=
mr
2
·
θ.
(3)
To
f
nd the
r
EOM we
f
rst calculate
∂L
∂r
=
mr
·
θ
2
−
k
r
,
(4)
wh
icha
l
lowsustowr
itetheEOMas
m
··
r
−
mr
·
θ
2
+
k
r
=0
.
(5)
Using Eq.(3) we can eliminate
·
θ
from Eq.(5) to obtain the equation of motion
for the equivalent onedimensional radial problem
m
··
r
=
f
0
(
r
)
.
(6)
where the e
f
ective force is
f
0
(
r
)=
−
k
r
+
l
2
mr
3
.
(7)
Note that the angular momentum
l
is a parameter whose value may be varied
by choosing di
f
erent initial values for
r
and
·
θ
.
To use the e
f
ective potential
V
0
to discuss the qualitative nature of the
motion, we need to make qualitative plots of
V
0
versus
r
for di
f
erent values of
l
.S
i
n
c
e
f
0
(
r
)=
−
dV
0
/dr
, we immediately realize that
V
0
(
r
)=
V
(
r
)+
l
2
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 Fall '10
 Wilemski
 mechanics, Force, circular orbit, Celestial mechanics, Elliptic orbit, Eqs.

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