Editing Laplace–Runge–Lenz vector (section)

== History of rediscovery ==
The LRL vector {{math|'''A'''}} is a constant of motion of the Kepler problem, and is useful in describing astronomical orbits, such as the motion of planets and binary stars. Nevertheless, it has never been well known among physicists, possibly because it is less intuitive than momentum and angular momentum. Consequently, it has been rediscovered independently several times over the last three centuries.<ref name="goldstein_1975_1976" />

[[Jakob Hermann]] was the first to show that {{math|'''A'''}} is conserved for a special case of the inverse-square central force,<ref>{{cite journal| last = Hermann | first = J. | author-link = Jakob Hermann | date = 1710 | title = Metodo d'investigare l'Orbite de' Pianeti, nell' ipotesi che le forze centrali o pure le gravità degli stessi Pianeti sono in ragione reciproca de' quadrati delle distanze, che i medesimi tengono dal Centro, a cui si dirigono le forze stesse | journal = Giornale de Letterati d'Italia | volume = 2 | pages = 447–467}}<br />{{cite journal| last = Hermann | first = J. | author-link = Jakob Hermann | date = 1710 | title = Extrait d'une lettre de M. Herman à M. Bernoulli datée de Padoüe le 12. Juillet 1710 | journal = Histoire de l'Académie Royale des Sciences | volume = 1732 | pages = 519–521}}</ref> and worked out its connection to the eccentricity of the orbital [[ellipse]]. Hermann's work was generalized to its modern form by [[Johann Bernoulli]] in 1710.<ref>{{cite journal| last = Bernoulli | first = J. | author-link = Johann Bernoulli | date = 1710 | title = Extrait de la Réponse de M. Bernoulli à M. Herman datée de Basle le 7. Octobre 1710 | journal = Histoire de l'Académie Royale des Sciences | volume = 1732 | pages = 521–544}}</ref> At the end of the century, Pierre-Simon de Laplace rediscovered the conservation of {{math|'''A'''}}, deriving it analytically, rather than geometrically.<ref>{{cite book | last = Laplace | first = P. S. | author-link = Laplace | date = 1799 | title = Traité de mécanique celeste | url = https://archive.org/details/traitdemcani02lapl | pages = Tome I, Premiere Partie, Livre II, pp.165ff | publisher = Paris, Duprat | no-pp = true}}</ref> In the middle of the nineteenth century, [[William Rowan Hamilton]] derived the equivalent eccentricity vector defined [[#Alternative scalings, symbols and formulations|below]],<ref name="hamilton_1847_quaternions" /> using it to show that the momentum vector {{math|'''p'''}} moves on a circle for motion under an inverse-square central force (Figure&nbsp;3).<ref name="hamilton_1847_hodograph" />

At the beginning of the twentieth century, [[Josiah Willard Gibbs]] derived the same vector by [[vector analysis]].<ref>{{cite book | last = Gibbs | first = J. W. | author-link = Josiah Willard Gibbs |author2=Wilson E. B. | date = 1901 | title = Vector Analysis | url = https://archive.org/details/vectoranalysisa01gibbgoog | publisher = Scribners | location = New York | page = [https://archive.org/details/vectoranalysisa01gibbgoog/page/n160 135]}}</ref> Gibbs' derivation was used as an example by Carl Runge in a popular [[Germany|German]] textbook on vectors,<ref>{{cite book | last = Runge | first = C. | author-link = Carl David Tolmé Runge | date = 1919 | title = Vektoranalysis | publisher = Hirzel | location = Leipzig | volume = I }}</ref> which was referenced by Wilhelm Lenz in his paper on the (old) quantum mechanical treatment of the hydrogen atom.<ref>{{cite journal | last = Lenz | first = W. | author-link = Wilhelm Lenz | date = 1924 | title = Über den Bewegungsverlauf und Quantenzustände der gestörten Keplerbewegung | journal = Zeitschrift für Physik | volume = 24 | issue = 1 | pages = 197–207 | doi = 10.1007/BF01327245|bibcode = 1924ZPhy...24..197L | s2cid = 121552327 }}</ref> In 1926, [[Wolfgang Pauli]] used the LRL vector to derive the energy levels of the hydrogen atom using the [[matrix mechanics]] formulation of quantum mechanics,<ref name="pauli_1926" /> after which it became known mainly as the ''Runge–Lenz vector''.<ref name="goldstein_1975_1976" />