Laplace–Runge–Lenz vector
Vector used in astronomy / From Wikipedia, the free encyclopedia
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In classical mechanics, the Laplace–Runge–Lenz (LRL) vector is a vector used chiefly to describe the shape and orientation of the orbit of one astronomical body around another, such as a binary star or a planet revolving around a star. For two bodies interacting by Newtonian gravity, the LRL vector is a constant of motion, meaning that it is the same no matter where it is calculated on the orbit;[1][2] equivalently, the LRL vector is said to be conserved. More generally, the LRL vector is conserved in all problems in which two bodies interact by a central force that varies as the inverse square of the distance between them; such problems are called Kepler problems.[3][4][5][6]
The hydrogen atom is a Kepler problem, since it comprises two charged particles interacting by Coulomb's law of electrostatics, another inverse-square central force. The LRL vector was essential in the first quantum mechanical derivation of the spectrum of the hydrogen atom,[7][8] before the development of the Schrödinger equation. However, this approach is rarely used today.
In classical and quantum mechanics, conserved quantities generally correspond to a symmetry of the system.[9] The conservation of the LRL vector corresponds to an unusual symmetry; the Kepler problem is mathematically equivalent to a particle moving freely on the surface of a four-dimensional (hyper-)sphere,[10] so that the whole problem is symmetric under certain rotations of the four-dimensional space.[11] This higher symmetry results from two properties of the Kepler problem: the velocity vector always moves in a perfect circle and, for a given total energy, all such velocity circles intersect each other in the same two points.[12]
The Laplace–Runge–Lenz vector is named after Pierre-Simon de Laplace, Carl Runge and Wilhelm Lenz. It is also known as the Laplace vector,[13][14] the Runge–Lenz vector[15] and the Lenz vector.[8] Ironically, none of those scientists discovered it.[15] The LRL vector has been re-discovered and re-formulated several times;[15] for example, it is equivalent to the dimensionless eccentricity vector of celestial mechanics.[2][14][16] Various generalizations of the LRL vector have been defined, which incorporate the effects of special relativity, electromagnetic fields and even different types of central forces.[17][18][19]