A force Fθ = mr˙θ˙
With the given force, Eq. (3.51) becomes
0 = m(r¨ − rθ˙2),
and
mr˙θ˙ = m(rθ¨ + 2r˙θ˙).
(3.142)
The second of these equations gives −r˙θ˙ = rθ¨. Therefore,
∫ θ¨/θ˙ dt = −∫ r˙/r dt ⇒ lnθ˙ = −ln r + C ⇒θ˙ = D/r, (3.143)
where D = eC is a constant of integration, determined by the initial conditions. Substituting this value of θ˙ into the first of Eqs. (3.142), and then multiplying through by r˙ and integrating, gives
r¨ = r(D/r)2 ⇒ ∫ r¨r˙ dt = D2 ∫ r˙/r dt ⇒ r˙2/2 = D2 ln r + E. (3.144)
Therefore,
r˙ = Alnr+B, (3.145)
where A ≡ 2D2 and B ≡ 2E.