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

where .

Suppose, for instance, that we wish to find the potential that causes an object to execute the spiral orbit

Substitution of into Equation (5.2) yields

(5.4) |

Integrating, we obtain

(5.5) |

or

(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.