Motion in a General Central Force-Field

Consider the motion of an object in a general (attractive) central force-field characterized by the potential energy per unit mass function $V(r)$. Since the force-field is central, it still remains true that

$$h = r^2\,\dot{\theta}$$ (370)

is a constant of the motion. As is easily demonstrated, Eq. (328) generalizes to

$$\frac{d^2 u}{d\theta^2} + u = - \frac{1}{h^2}\frac{dV}{du},$$ (371)

where $u=r^{-1}$.

Suppose, for instance, that we wish to find the potential $V(r)$ which causes an object to execute the spiral orbit

$$r = r_0\,\theta^{\,2}.$$ (372)

Substitution of $u = (r_0\,\theta^2)^{-1}$ into Eq. (371) yields

$$\frac{d V}{du} = - h^2\left(6\,r_0\,u^2 + u\right).$$ (373)

Integrating, we obtain

$$V(u) = -h^2\left(2\,r_0\,u^3 + \frac{u^2}{2}\right),$$ (374)

or

$$V(r) = - h^2\left(\frac{2\,r_0}{r^3} + \frac{1}{2\,r^2}\right).$$ (375)

In other words, the spiral pattern (372) is obtained from a mixture of an inverse-square and inverse-cube potential.