Kepler and Oscillator Elliptical Orbits
According to Bertrand's theorem, the only central forces that result in closed orbits for all bound particles, independent of initial conditions, are the inverse-square law or Hooke's law. The corresponding central forces are proportional to or , associated with the Kepler problem or the isotropic two-dimensional oscillator, respectively. Isaac Newton recognized in the Principia  that both problems could admit elliptical orbits. For the oscillator, the center of force is at the center of the ellipse, while for the Kepler problem, it is at one focus of the ellipse.[more]
An ellipse with semimajor axis and semiminor axis can be represented by the equation
The eccentricity of the ellipse is defined by
so that . For a circular orbit, , while larger values of give a flatter ellipse. The ellipse can be tilted by an angle from the horizontal. The center of force is represented by a black dot, the orbiting body (perhaps a planet) by a red dot. The red arrow shows the attractive central force maintaining the orbit.
Animating the graphic shows both the oscillator and Kepler orbits for the selected values of the parameters.[less]
The motion of the oscillator is given by
An elliptical Kepler orbit can be represented in polar coordinates by
where the energy and angular momentum of the orbit are given by
Here is the mass of the center of force, the mass of the orbiting body and the gravitational constant. It is assumed that (otherwise reduced mass must be introduced). The period of the orbit is given by Kepler's third law:
For simplicity, assume units such that .
The mean anomaly (not to be confused with the mass above) is the hypothetical angle about the center of the Kepler ellipse assuming uniform motion. Thus
The actual motion is described by the eccentric anomaly , related to by Kepler's equation
The equation is transcendental, but an accurate solution for in the form of a Fourier sine series can be derived :
 S. Chandrasekhar, Newton's Principia for the Common Reader, Oxford: Clarendon Press, 1995 pp. 114–125.
 M. A. Murison. "Series Solutions of Kepler's Equation." (Nov 30, 2020) www.murison.alpheratz.net/Maple/KeplerSolve/KeplerSolve.pdf.