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

(5.1) 
is a constant of the motion. (See Section 4.5.) As is easily demonstrated, Equation (4.28)
generalizes to

(5.2) 
where
.
Suppose, for instance, that we wish to find the potential that causes
an object to execute the spiral orbit

(5.3) 
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 inversesquare and inversecube potential.