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The Runge-Lenz Vector

The Runge–Lenz (RL) vector, also commonly known as the Laplace–Runge–Lenz vector, is a "hidden" constant of the motion for both the classical Kepler and quantum Coulomb problems. In classical mechanics, it implies that the energy of planetary motion depends only on the semimajor axis of an elliptical orbit and is independent of the angular momentum. In quantum mechanics, it accounts for the 2-2, 3-3-3, and so on orbital degeneracy in the nonrelativistic hydrogen atom. As noted by Goldstein [2] and [3], this vector was actually discovered independently over the years by several other scientists, including possibly Newton himself.
The RL vector can be derived starting with Newton's second law applied to gravitational attraction, written in the form . Now take the cross product of both sides with the orbital angular momentum , noting that is a constant of the motion. This gives . Using the vector identities and , we find or , showing that , defined as the Runge–Lenz vector, is a constant of the motion. Note that so that the RL vector is normal to the orbital angular momentum and therefore in the plane of the planetary orbit.
The equation of the orbit is found easily by evaluating the scalar product , which gives the equation of a conic section in plane polar coordinates: , where is the eccentricity, related to the RL vector magnitude by . The origin of the coordinate system is taken as the location of the heavy particle (). The perihelion of the orbit (the distance of closest approach) is given by , corresponding to .
The energy of an orbiting planet is given by . The orbit is an ellipse when with ( for a circular orbit), a parabola when and , and a hyperbola when and .
In this Demonstration, orbits for input values of and are shown. For simplicity, we set , thus . If you trigger the orbital motion, you will see the periodic orbit of an ellipse or circle. For a parabola or hyperbola, there is just a single pass of the planetary body, but this is shown repeatedly.

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References
[1] H. Goldstein, Classical Mechanics, 2nd ed., Reading, MA: Addison Wesley, 1980, pp. 102–105, 421–422.
[2] H. Goldstein, "Prehistory of the Runge–Lenz Vector," American Journal of Physics, 43, 1975 pp. 735–738.
[3] H. Goldstein, "More on the Prehistory of the Runge–Lenz Vector," American Journal of Physics, 44, 1976 pp. 1123–1124.
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