Laplace-Runge-Lenz vector
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In classical mechanics, for a central force with <math>1/r\!\,<math> potential, the Laplace-Runge-Lenz vector is a conserved vector of motion. It is defined as:
- <math> \mathbf{A}=\mathbf{p} \times \mathbf{L} - m k \frac{\mathbf{r}}{r}<math>
where:
- <math>\mathbf{r}\!\,<math> is the position vector of the mass <math>m\!\,<math>,
- <math>\mathbf{L}\!\,<math> is the angular momentum,
- <math>k\!\,<math> is a parameter that describes strength of the potential.
The vector is based at the center of attraction and points toward the pericenter. The magnitude for a periodic orbit with eccentricity <math>e\!\,<math> is given by:
- <math> |\mathbf{A}|= m k e <math>
The quantities <math>A\!\,<math>, the angular momentum <math>L\!\,<math> and the energy <math>E\!\,<math> are not all independent. One can show that the magnitude of the Laplace-Runge-Lenz vector is:
- <math> A^2= m^2 k^2 +2 m E L^2 \!\,<math>
Furthermore, from the definition of the vector, we can easily see that:
- <math> \mathbf{A} \cdot \mathbf{L}= 0 <math>,
Therefore, these two constraints show that the seven constants of motion (3+3+1) are not independent; but there are five independent constants of motion out of the seven.
The vector was originally discovered by Laplace in the late 18th century in the context of celestial mechanics, so that it was known as Laplace vector, but was rediscovered in the early 20th century in the context of quantum mechanics as the Runge-Lenz vector.
The Laplace-Runge-Lenz vector is a geometric property of an elliptic orbit generated by a centripetal force, so it occurs both in the orbit of a planet around the sun and in the orbit of an electron around a proton, even though different forces in different scales are involved. This makes LRL vector relevant for both macro (e.g. celestial mechanics, astrodynamics) and micro scale (e.g. quantum mechanics). Also, the LRL vector correlates positively with the eccentricity of the orbit.
The hodograph of an elliptic orbit is always a circle. Points on the hodograph are traced by the head of the velocity vector as it changes with time. The tail of the velocity vector is a constant point, the origin (of the hodograph), within the circle but not necessarily the center. Then, the vector which joins the center of the circle to the origin is the LRL vector.
See also
External links
- Mysteries of the gravitational 2-body problem (http://math.ucr.edu/home/baez/gravitational.html), by John Baez