Kepler problem

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 "F" that varies in strength as the inverse square of the distance "r" between them. The force may be either attractive or repulsive. The problem is to find the position or speed of the two bodies over time given their masses, positions, and velocities. Using classical mechanics, the solution can be expressed as a Kepler orbit using six orbital elements.
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").
For a discussion of the Kepler problem specific to radial orbits, see Radial trajectory. General relativity provides more accurate solutions to the two-body problem, especially in strong gravitational fields.
Applications.
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. Examples include the hydrogen atom, positronium and muonium, which have all played important roles as model systems for testing physical theories and measuring constants of nature.
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 has often been used to develop new methods in classical mechanics, such as Lagrangian mechanics, Hamiltonian mechanics, the Hamilton–Jacobi equation, and action-angle coordinates. 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.
Mathematical definition.
The central force F that varies in strength as the inverse square of the distance "r" between them:
where "k" is a constant and formula_2 represents the unit vector along the line between them. The force may be either attractive ("k"<0) or repulsive ("k">0). The corresponding scalar potential (the potential energy of the non-central body) is:
Solution of the Kepler problem.
The equation of motion for the radius formula_4 of a particle 
of mass formula_5 moving in a central potential formula_6 is given by Lagrange's equations
If "L" is not zero the definition of angular momentum allows a change of independent variable from formula_12 to formula_13
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 formula_17 and multiplying both sides by formula_18
After substitution and rearrangement:
For an inverse-square force law such as the gravitational or electrostatic potential, the potential can be written
The orbit formula_23 can be derived from the general equation
whose solution is the constant formula_25 plus a simple sinusoid
where formula_27 (the eccentricity) and formula_28 (the phase offset) are constants of integration.
This is the general formula for a conic section that has one focus at the origin; formula_29 corresponds to a circle, formula_30 corresponds to an ellipse, formula_31 corresponds to a parabola, and formula_32 corresponds to a hyperbola. The eccentricity formula_27 is related to the total energy formula_34 (cf. the Laplace–Runge–Lenz vector)
Comparing these formulae shows that formula_36 corresponds to an ellipse (all solutions which are closed orbits are ellipses), formula_37 corresponds to a parabola, and formula_38 corresponds to a hyperbola. In particular, formula_39 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.
Solution in pedal coordinates.
If we restrict ourselves to the orbiting plane, there is an easy way how to obtain a rough shape of the orbit (without the information about parametrization) in pedal coordinates. Remember that a given point formula_40 on a curve in pedal coordinates is given by two numbers formula_41, where formula_42 is the distance from the origin and formula_43 is the distance of the origin to the tangent line at formula_40 (the symbol formula_45 stands for a vector perpendicular to formula_46—exact orientation is unimportant here).
The Kepler problem in a plane asks for a solution of the system of differential equations:
where formula_48 is the product of the gravitational body's mass and gravitational constant. Making the scalar product of the equation with formula_49 we obtain