Math 2280 - Explanation for Step 6
Dylan Zwick
Spring 2009
Class, there has been some confusion about how we handle step 6 of
the project, as it requires some chain rule jiu jitsu. So, here’s an explanation
of how it works.
If we define r = 1/z then the chain rule tells us that:
dr
dr dz
1 dz
=
=− 2 .
dt
dz dt
z dt
If we then use the relation from the textbook:
r2
dθ
=h
dt
we get:
−
1 dz
dt dz
dz
dz
= −r 2
= −h
= −h .
2
z dt
dt
dθ dt
dθ
And so we derive the relation:
dz
dr
= −h .
dt
dθ
If we differentiate again with respect to t and again use the chain rule
we get:
1
d2 z dθ
d2 r
=
−h
.
dt2
dθ2 dt
Now, if we agin use our relation:
r2
dθ
=h
dt
then we get:
d2 z h
h2 d2 z
d2 r
=
−h
=
−
.
dt2
dθ2 r 2
r 2 dθ2
If we then equate this with our relation from the textbook:
d2 r h2
k
− 3 =− 2
2
dt
r
r
we get:
−
k
h2 d2 z h2
− 3 =− 2
2
2
r dθ
r
r
which simplifies to
d2 z 1
k
+ = 2.
2
dθ
r
h
If we then use our defining relation for z, namely z = 1/r, then we get
the relation:
k
d2 z
+ z = 2.
2
dθ
h
Which is what we want to derive.
2