where xis a fixed vector of integration. We define the Runge–Lenz vector:
A:= x
k=˙q×J
k−ˆq.
4. Let’s expand ˙q×Jusing the vector identity given in Exercise 1:
˙q×J=m[ ˙q×(q×˙q)] = m[|˙q|2q−(q·˙q) ˙q].
Thus it follows that
( ˙q×J)·q=m[|˙q|2|q|2−(q·˙q)2]
and if neither qnor ˙qis zero we have that
( ˙q×J)·q=m|˙q|2|q|2[1 −cos2ϕ] = m|˙q|2|q|2sin2ϕ
where ϕis the angle between qand ˙q. This last expression is exactly J·J/m so that we have shown:
( ˙q×J)·q=J·J/m (6)
(if either qor ˙qis zero, then Jis zero, and the above expression remains valid ). Combining (6)
with the definition of Ayields:
A·q=J·J
km − |q|.(7)
5. If we write A·qas |A||q|cos θwhere θis the angle between Aand qwe see that (7) gives:
|A||q|cos θ=J·J
km − |q|
which can be solved for |q|to yield:
|q|=J·J
km
1
1 + |A|cos θ.(8)
2
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