Editing Laplace–Runge–Lenz vector (section)

==Context==
A single particle moving under any [[conservation of energy|conservative]] central force has at least four constants of motion: the total energy {{mvar|E}} and the three [[Cartesian coordinate system|Cartesian components]] of the [[angular momentum]] vector {{math|'''L'''}} with respect to the center of force.<ref name="goldstein_1980d">{{cite book | last=Goldstein | first=H. | author-link=Herbert Goldstein | date=1980 | title=Classical Mechanics | edition=2nd | publisher=Addison Wesley | pages=1–11}}</ref><ref name="symon_1971">{{cite book | last=Symon | first=K. R. | author-link=Keith Symon | date=1971 | title=Mechanics | edition=3rd | publisher=Addison Wesley | pages=103–109, 115–128}}</ref> The particle's orbit is confined to the plane defined by the particle's initial [[momentum]] {{math|'''p'''}} (or, equivalently, its [[velocity]] {{math|'''v'''}}) and the vector {{math|'''r'''}} between the particle and the center of force<ref name="goldstein_1980d" /><ref name="symon_1971" /> (see Figure 1). This plane of motion is perpendicular to the constant angular momentum vector {{math|1='''L''' = '''r''' × '''p'''}}; this may be expressed mathematically by the vector [[dot product]] equation {{math|1='''r''' ⋅ '''L''' = 0}}. Given its [[#Mathematical definition|mathematical definition]] below, the Laplace–Runge–Lenz vector (LRL vector) {{math|'''A'''}} is always perpendicular to the constant angular momentum vector {{math|'''L'''}} for all central forces ({{math|1='''A''' ⋅ '''L''' = 0}}). Therefore, {{math|'''A'''}} always lies in the plane of motion. As shown [[#Derivation of the Kepler orbits|below]], {{math|'''A'''}} points from the center of force to the [[periapsis]] of the motion, the point of closest approach, and its length is proportional to the eccentricity of the orbit.<ref name="goldstein_1980" />

The LRL vector {{math|'''A'''}} is constant in length and direction, but only for an inverse-square central force.<ref name="goldstein_1980" /> For other [[classical central-force problem|central forces]], the vector {{math|'''A'''}} is not constant, but changes in both length and direction. If the central force is ''approximately'' an inverse-square law, the vector {{math|'''A'''}} is approximately constant in length, but slowly rotates its direction.<ref name="arnold_1989b" /> A ''generalized'' conserved LRL vector <math>\mathcal{A}</math> [[#Generalizations to other potentials and relativity|can be defined]] for all central forces, but this generalized vector is a complicated function of position, and usually not [[expressible in closed form]].<ref name="fradkin_1967" /><ref name="yoshida_1987" />

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 mechanics|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 bracket]]s, 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.