Physics 325
Lecture 8
Motion in a Central Field (cont.)
Recall from our discussing in Lecture 7 that it is sometimes convenient to study the
motion of a particle in an isotropic central field
ˆ
( )
F
f r r
by directly solving the
differential equations
from Newton’s 2
nd
Law. We arrived at a differential equation for
1/
u
r
given by Equation 7.8:
2
1
2
2
2
(
)
d u
m
u
f u
d
L u
This is the differential equation of the orbit of a particle moving under the influence of a
central field.
Conversely, if one is given the orbit
1
( )
r
r
u
, then one can obtain the force function
by differentiating the orbit to get
2
2
d u d
and inserting this into the above differential
equation.
Example:
A particle in a central field moves in a spiral orbit given by
2
r
c
Determine the force function
( )
f r
and
( )
t
.
We have
2
1
u
c
and
2
3
4
2
2
2
6
6
du
d u
cu
d
c
d
c
From Equation 7.8,
2
1
2
2
6
(
)
m
cu
u
f u
L u
Hence,
2
1
4
3
(
)
(6
)
L
f u
cu
u
m
and
2
4
3
6
1
( )
L
c
f r
m
r
r
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To find
(
t
)
, we can use the fact that
2
constant
r
L m
:
2
2
4
Lu
L
m
mc
or
4
2
L
d
dt
mc
By integrating, we find
5
2
5
L
t
mc
where the constant of integration has been taken to be zero so that
0
at
t
0
.
Finally, we can write
1/5
2
5
( )
L
t
t
mc
Gravitational Force and Potential
Now we return specifically to the problem of gravity and planetary motion.
First with
some general considerations regarding gravitational forces, and then we will turn to the
question of planetary orbits and Kepler’s laws.
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 Spring '08
 Staff
 Physics, mechanics, Mass, Potential Energy, General Relativity, Gravitational Force, Gravitational forces

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