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' to be solved is to find the position or speed of the two bodies over time given their masses and initial positions and velocities. Using classical mechanics, the solution can be expressed as a Kepler orbit using six orbital elements. 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' to be solved is to find the position or speed of the two bodies over time given their masses and initial 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 solve 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. The Kepler problem in general relativity produces more accurate predictions, especially in strong gravitational fields. 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. The central force F that varies in strength as the inverse square of the distance r between them: where k is a constant and r ^ {displaystyle mathbf {hat {r}} } 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: The equation of motion for the radius r {displaystyle r} of a particle of mass m {displaystyle m} moving in a central potential V ( r ) {displaystyle V(r)} is given by Lagrange's equations If L is not zero the definition of angular momentum allows a change of independent variable from t {displaystyle t} to θ {displaystyle heta }