
165Motion Under a Central Force
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We note that Equation 6.27 is Newton’s second law written in the form of equations of motion. It
is more convenient to use the new variable u (= 1/r). In terms of u, we have
θ
µ
µ==
l
r
lu
2
2
/
(6.28)
and
d
d
d
d
d
r
u
u
u
u
l
u
==−=−=−
−− −
1
22 1
θµ
.
(6.29)
If we differentiate dr/dt once more, we obtain
d
d
d
d
d
d
d
d
2
2
2
2
2
r
t
l
t
ululu
=−
=− =−
µθµ
θ
θ
µ
dd
d
2
2
u
θ
. (6.30)
Substituting Equations 6.28 through 6.30 into Equation 6.27, we obtain
d
d
/
2
222
1
1
u
u
lu
µ
+=−
. (6.31)
This equation is particularly useful if we wish to nd the force law when the equation of orbit
r=r(θ) is given. To illustrate ...