The Kepler problem arises in many contexts, some beyond the physics studied by Kepler himself. The Kepler problem is important in celestial mechanics, since Newtonian gravity obeys an inverse square law. Examples include a satellite moving about a planet, a planet about its sun, or two binary stars about each other. 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.
The Kepler problem and the simple harmonic oscillator problem are the two most fundamental problems in classical mechanics. 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). 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 equation of motion for the radius of a particle
of mass moving in a central potential is given by Lagrange's equations
and the angular momentum is conserved. For illustration, the first term on the left-hand side is zero for circular orbits, and the applied inwards force equals the centripetal force requirement , as expected.
If L is not zero the definition of angular momentum allows a change of independent variable from to
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 and multiplying both sides by
After substitution and rearrangement:
For an inverse-square force law such as the gravitational or electrostatic potential, the potential can be written
The orbit can be derived from the general equation
whose solution is the constant plus a simple sinusoid
where (the eccentricity) and (the phase offset) are constants of integration.
This is the general formula for a conic section that has one focus at the origin; corresponds to a circle, corresponds to an ellipse, corresponds to a parabola, and corresponds to a hyperbola. The eccentricity is related to the total energy (cf. the Laplace–Runge–Lenz vector)
Comparing these formulae shows that corresponds to an ellipse (all solutions which are closed orbits are ellipses), corresponds to a parabola, and corresponds to a hyperbola. In particular, for perfectly circular orbits (the central force exactly equals the centripetal force requirement, which determines the required angular velocity for a given circular radius).
For a repulsive force (k > 0) only e > 1 applies.