In classical mechanics, the Kepler problem is a special case of the two-body problem, in which the two bodies interact by a central force that varies in strength as the inverse square of the distance between them.
The Kepler problem is named after Johannes Kepler, who proposed Kepler's laws of planetary motion (which are part of classical mechanics and solved the problem for the orbits of the planets) and investigated the types of forces that would result in orbits obeying those laws (called Kepler's inverse problem).
General relativity provides more accurate solutions to the two-body problem, especially in strong gravitational fields.
[1]: 92 The Kepler problem is important in celestial mechanics, since Newtonian gravity obeys an inverse square law.
The Kepler problem is also important in the motion of two charged particles, since Coulomb’s law of electrostatics also obeys an inverse square law.
They are the only two problems that have closed orbits for every possible set of initial conditions, i.e., return to their starting point with the same velocity (Bertrand's theorem).
[1]: 92 The Kepler problem also conserves the Laplace–Runge–Lenz vector, which has since been generalized to include other interactions.
The solution of the Kepler problem allowed scientists to show that planetary motion could be explained entirely by classical mechanics and Newton’s law of gravity; the scientific explanation of planetary motion played an important role in ushering in the Enlightenment.
The Kepler problem begins with the empirical results of Johannes Kepler arduously derived by analysis of the astronomical observations of Tycho Brache.
After some 70 attempts to match the data to circular orbits, Kepler hit upon the idea of the elliptic orbit.
He eventually summarized his results in the form of three laws of planetary motion.
[2] What is now called the Kepler problem was first discussed by Isaac Newton as a major part of his Principia.
His "Theorema I" begins with the first two of his three axioms or laws of motion and results in Kepler's second law of planetary motion.
Next Newton proves his "Theorema II" which shows that if Kepler's second law results, then the force involved must be along the line between the two bodies.
In other words, Newton proves what today might be called the "inverse Kepler problem": the orbit characteristics require the force to depend on the inverse square of the distance.
[3]: 107 The central force F between two objects varies in strength as the inverse square of the distance r between them: where k is a constant and
represents the unit vector along the line between them.
The corresponding scalar potential is: The equation of motion for the radius
For illustration, the first term on the left-hand side is zero for circular orbits, and the applied inwards force
If L is not zero the definition of angular momentum allows a change of independent variable from
giving the new equation of motion that is independent of time The expansion of the first term is This equation becomes quasilinear on making the change of variables
After substitution and rearrangement: For an inverse-square force law such as the gravitational or electrostatic potential, the scalar potential can be written The orbit
can be derived from the general equation whose solution is the constant
This is the general formula for a conic section that has one focus at the origin;
for perfectly circular orbits (the central force exactly equals the centripetal force requirement, which determines the required angular velocity for a given circular radius).