Binet equation
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. Equation. The shape of an orbit is often conveniently described in terms of relative distance formula_1 as a function of angle formula_2 . For  the Binet equation, the orbital shape is instead more concisely described by the reciprocal formula_3 as a function of formula_2. Define the specific angular momentum as formula_5 where formula_6 is the angular momentum and formula_7 is the mass. The Binet equation, derived in the next section, gives the force in terms of the function formula_8: Derivation. Newton's Second Law for a purely central force is The conservation of angular momentum requires that Derivatives of formula_1 with respect to time may be rewritten as derivatives of formula_3 with respect to angle: Combining all of the above, we arrive at The  general solution is formula_16 where formula_17 is the initial coordinate of the particle. Examples. Kepler problem. Classical. 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 If the angle formula_2 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 formula_22 the semi-latus rectum (equal to formula_23) and formula_24 the orbital eccentricity. Relativistic. The relativistic equation derived for Schwarzschild coordinates is where formula_26 is  the speed of light and formula_27 is the Schwarzschild radius. And for Reissner–Nordström metric we will obtain where formula_29 is the electric charge and formula_30 is the vacuum permittivity. Inverse Kepler problem. Consider the inverse Kepler problem. 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 The force law is therefore which is the anticipated inverse square law. Matching the orbital formula_33 to physical values like formula_34 or formula_35 reproduces Newton's law of universal  gravitation or Coulomb's law, respectively. The effective force for Schwarzschild coordinates is 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). In the parameterized post-Newtonian formalism we will obtain where formula_38 for the general relativity and formula_39 in the classical case. Cotes spirals. An inverse cube force law has the form The shapes of the orbits of an inverse cube law are known as Cotes spirals. The Binet equation shows that the orbits must be solutions to the equation  The differential equation has three kinds of solutions, in analogy to the different conic sections of the Kepler problem. When formula_42, the solution is the epispiral, including the pathological case of a straight line when formula_43. When formula_44, the solution is the hyperbolic spiral. When formula_45 the solution is Poinsot's spiral. Off-axis circular motion. 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. 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 formula_46 is Differentiating formula_48 twice and making use of the Pythagorean identity gives The force law is thus Note that solving the general inverse problem, i.e. constructing the orbits of an attractive formula_51 force law, is a considerably more difficult problem because it is equivalent to solving which is a second order nonlinear differential equation.