This is known as the Binet equation.

For the Binet equation, the orbital shape is instead more concisely described by the reciprocal as a function of .

For momentum , and their ratio the equation of motion is (see Binet equation)

Solving the above differential equation with respect to time(See also Binet equation) yields:

This equation becomes quasilinear on making the change of variables and multiplying both sides by (see also Binet equation)

The Binet equation for u(φ) can be solved numerically for nearly any central force F(1/u).

The Binet equation, derived by Jacques Philippe Marie Binet, provides the form of a central force given the shape of the orbital motion in plane polar coordinates.

Although the Binet equation fails to give a unique force law for circular motion about the center of force, the equation can provide a force law when the circle's center and the center of force do not coincide.