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In classical mechanics, the Laplace–Runge–Lenz vector (or simply the 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 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; 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. 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, 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. 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, so that the whole problem is symmetric under certain rotations of the four-dimensional space. 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. 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, the Runge–Lenz vector and the Lenz vector. Ironically, none of those scientists discovered it. The LRL vector has been re-discovered several times〔 〕 and is also equivalent to the dimensionless eccentricity vector of celestial mechanics. 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. ==Context== A single particle moving under any conservative central force has at least four constants of motion, the total energy ''E'' and the three Cartesian components of the angular momentum vector L with respect to the origin. The particle's orbit is confined to a plane defined by the particle's initial momentum p (or, equivalently, its velocity v) and the vector r between the particle and the center of force (see Figure 1, below). As defined below (see Mathematical definition), the Laplace–Runge–Lenz vector (LRL vector) A always lies in the plane of motion for any central force. However, A is constant only for an inverse-square central force.〔 For most central forces, however, this vector A is not constant, but changes in both length and direction; if the central force is ''approximately'' an inverse-square law, the vector A is approximately constant in length, but slowly rotates its direction. A ''generalized'' conserved LRL vector can be defined for all central forces, but this generalized vector is a complicated function of position, and usually not expressible in closed form. 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. The LRL vector differs from other conserved quantities in the following property. Whereas for typical conserved quantities, there is a corresponding cyclic coordinate in the three-dimensional Lagrangian of the system, there does ''not'' exist such a coordinate for the LRL vector. Thus, the conservation of the LRL vector must be derived directly, e.g., by the method of Poisson brackets, as described below. Conserved quantities of this kind are called "dynamic", in contrast to the usual "geometric" conservation laws, e.g., that of the angular momentum. 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「Laplace–Runge–Lenz vector」の詳細全文を読む スポンサード リンク
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