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Next: Motion in nearly circular Up: Orbits in central force Previous: Introduction

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

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

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

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

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

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

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

Integrating, we obtain

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

or

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


next up previous
Next: Motion in nearly circular Up: Orbits in central force Previous: Introduction
Richard Fitzpatrick 2016-03-31