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.
The equation can also be used to derive the shape of the orbit for a given force law, but this usually involves the solution to a second order nonlinear, ordinary differential equation.
A unique solution is impossible in the case of circular motion about the center of force.
The shape of an orbit is often conveniently described in terms of relative distance
For the Binet equation, the orbital shape is instead more concisely described by the reciprocal
Define the specific angular momentum as
The Binet equation, derived in the next section, gives the force in terms of the function
Newton's Second Law for a purely central force is
The conservation of angular momentum requires that
with respect to time may be rewritten as derivatives of
The traditional Kepler problem of calculating the orbit of an inverse square law may be read off from the Binet equation as the solution to the differential equation
is measured from the periapsis, then the general solution for the orbit expressed in (reciprocal) polar coordinates is
The above polar equation describes conic sections, with
The relativistic equation derived for Schwarzschild coordinates is[2]
What kind of force law produces a noncircular elliptical orbit (or more generally a noncircular conic section) around a focus of the ellipse?
Differentiating twice the above polar equation for an ellipse gives
which is the anticipated inverse square law.
The effective force for Schwarzschild coordinates is[3]
where the second term is an inverse-quartic force corresponding to quadrupole effects such as the angular shift of periapsis (It can be also obtained via retarded potentials[4]).
In the parameterized post-Newtonian formalism we will obtain
An inverse cube force law has the form
The shapes of the orbits of an inverse cube law are known as Cotes spirals.
The differential equation has three kinds of solutions, in analogy to the different conic sections of the Kepler problem.
, the solution is the epispiral, including the pathological case of a straight line when
Consider for example a circular orbit that passes directly through the center of force.
A (reciprocal) polar equation for such a circular orbit of diameter
twice and making use of the Pythagorean identity gives
Note that solving the general inverse problem, i.e. constructing the orbits of an attractive
force law, is a considerably more difficult problem because it is equivalent to solving
which is a second order nonlinear differential equation.