Motion in 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 $V(r)$. Because the force field is central, it still remains true that

$$h = r^{\,2}\,\skew{5}\dot{\theta}$$ (5.1)

is a constant of the motion. (See Section 4.5.) As is easily demonstrated, Equation (4.28) generalizes to

$$\frac{d^{\,2} u}{d\theta^{\,2}} + u = - \frac{1}{h^{\,2}}\frac{dV}{du},$$ (5.2)

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

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

$$r = r_0\,\theta^{\,2}.$$ (5.3)

Substitution of $u = (r_0\,\theta^{\,2})^{-1}$ into Equation (5.2) yields

$$\frac{d V}{du} = - h^{\,2}\left(6\,r_0\,u^{\,2} + u\right).$$ (5.4)

Integrating, we obtain

$$V(u) = -h^{\,2}\left(2\,r_0\,u^{\,3} + \frac{u^{\,2}}{2}\right),$$ (5.5)

or

$$V(r) = - h^{\,2}\left(\frac{2\,r_0}{r^{\,3}} + \frac{1}{2\,r^{\,2}}\right).$$ (5.6)

In other words, the orbit specified by Equation (5.3) is obtained from a mixture of an inverse-square and inverse-cube potential.