In classical mechanics, the Laplace–Runge–Lenz vector is a constant of motion. For two bodies interacting by Newtonian gravity, the LRL vector is said to be conserved. It 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. The vector was essential in the first quantum mechanical derivation of the spectrum of the hydrogen atom.
About Laplace–Runge–Lenz vector in brief
In classical mechanics, the Laplace–Runge–Lenz vector is a constant of motion. For two bodies interacting by Newtonian gravity, the LRL vector is said to be conserved. It 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. The vector was essential in the first quantum mechanical derivation of the spectrum of the hydrogen atom, before the development of the Schrödinger equation. It has never been well-known among physicists, possibly because it is less intuitive than momentum and angular momentum. At the end of the 20th century, Pierre-Simon de Laplace discovered the conservation of A, and worked out its connection to the eccentricity of the orbital ellipse. It was rediscovered several times over the last three centuries, and was generalized to its modern form by Johann Bernoulli in 1710. The conservation of the L RL vector must be derived directly, e. g. by the method of Poisson brackets, as described below. It can be defined for all central forces, but it is a complicated function of position, and usually not expressible in closed form. LRL has been re-discovered several times and is also equivalent to the dimensionless eccentricity vector of celestial mechanics. In classical and quantum mechanics, conserved quantities generally correspond to a symmetry of the system. Whereas for typical Conserved quantities, there is a corresponding cyclic coordinate in the three-dimensional Lagrangian of a system, there does not exist such a coordinate for the LRC vector.
The LRL is useful in describing the motion of astronomical orbits, such as planets revolving around a star, because it describes the shape and orientation of the orbit of one astronomical body around another, like a planet revolvingaround a star. It’s also used in the theory of the universe’s rotation. The Laplace- runge-lenz vector is also known as the La place vector, the Runge–lenz vector and the Lenz vector, and it is also used to describe the movement of galaxies and stars in the Milky Way galaxy, among other things. For most central force, however, this vector A is not constant, but changes in both length and direction; if the central force is approximately an inverse-square law, A is approximately constant in length, but slowly rotates its direction. The plane of motion is perpendicular to the angular momentum vector L, which is constant; this may be expressed mathematically by the vector dot product equation r ⋅ L = 0; likewise, since A lies in that plane, A ⋉ L =0. A generalized conserved LRL Vector A can bedefined for allCentral forces, but this generalized vector is not expressable in closedform, and it is usually not expressedible inclosed form. For a special case of the central-square force, the first to show that A is Conserved is Jakob Hermann de La place.
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