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Laplace–Runge–Lenz vector
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{{short description|Vector used in astronomy}} {{hatnote|Throughout this article, vectors and their magnitudes are indicated by boldface and italic type, respectively. For example, <math>\left| \mathbf{A} \right| = A</math>.}} In [[classical mechanics]], the '''Laplace–Runge–Lenz vector''' ('''LRL vector''') is a [[vector (geometric)|vector]] used chiefly to describe the shape and orientation of the [[orbit (celestial mechanics)|orbit]] of one [[astronomical body]] around another, such as a [[binary star]] or a planet revolving around a star. For [[Two-body problem|two bodies interacting]] by [[Newton's law of universal gravitation|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;<ref name="goldstein_1980">{{cite book | last=Goldstein | first=H. | author-link=Herbert Goldstein | date=1980 | title=Classical Mechanics | edition=2nd | publisher=Addison Wesley | pages=102–105, 421–422}}</ref><ref name="taff_1985">{{cite book | last = Taff | first = L. G. | author-link = Laurence G. Taff | date = 1985 | title = Celestial Mechanics: A Computational Guide for the Practitioner | publisher = John Wiley and Sons | location = New York | pages = 42–43}}</ref> equivalently, the LRL vector is said to be ''[[Conservation law|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 law|inverse square]] of the distance between them; such problems are called [[Kepler problem]]s.<ref name="goldstein_1980b">{{cite book | last=Goldstein | first=H. | author-link=Herbert Goldstein | date=1980 | title=Classical Mechanics | edition=2nd | publisher=Addison Wesley | pages=94–102}}</ref><ref name="arnold_1989">{{cite book | last = Arnold | first = V. I. | author-link = Vladimir Arnold | date = 1989 | title = Mathematical Methods of Classical Mechanics | edition = 2nd | publisher = Springer-Verlag | location = New York | page = [https://archive.org/details/mathematicalmeth0000arno/page/38 38] | isbn = 0-387-96890-3 | url = https://archive.org/details/mathematicalmeth0000arno/page/38 }}</ref><ref name="sommerfeld_1989">{{cite book | last = Sommerfeld | first = A. | author-link = Arnold Sommerfeld | date = 1964 | title = Mechanics | series=Lectures on Theoretical Physics | volume = 1 | edition = 4th | translator = Martin O. Stern | publisher = Academic Press | location = New York | pages = 38–45}}</ref><ref name="lanczos_1970">{{cite book | last = Lanczos | first = C. | author-link = Cornelius Lanczos | date = 1970 | title = The Variational Principles of Mechanics | edition = 4th | publisher = Dover Publications | location = New York | pages = 118, 129, 242, 248 }}</ref> Thus 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 mechanic]]al derivation of the [[atomic emission spectrum|spectrum]] of the hydrogen atom,<ref name="pauli_1926">{{cite journal | last = Pauli | first = W. | author-link = Wolfgang Pauli | date = 1926 | title = Über das Wasserstoffspektrum vom Standpunkt der neuen Quantenmechanik | journal = Zeitschrift für Physik | volume = 36 | issue = 5 | pages = 336–363 | doi = 10.1007/BF01450175 | bibcode = 1926ZPhy...36..336P | s2cid = 128132824 }}</ref><ref name="bohm_1993">{{cite book | last = Bohm | first = A. | date = 1993 | title = Quantum Mechanics: Foundations and Applications | edition = 3rd | publisher = Springer-Verlag | location = New York | pages = 205–222}}</ref> 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 (physics)#Conservation laws and symmetry|symmetry]] of the system.<ref name="hanca_et_al_2004">{{cite journal |author1=Hanca, J. |author2=Tulejab, S. |author3=Hancova, M. |title=Symmetries and conservation laws: Consequences of Noether's theorem |journal=American Journal of Physics |volume=72 |issue=4 |pages=428–35 |year=2004 | doi = 10.1119/1.1591764 | url=http://www.eftaylor.com/pub/symmetry.html | bibcode = 2004AmJPh..72..428H }}</ref> The conservation of the LRL vector corresponds to an unusual symmetry; the Kepler problem is mathematically equivalent to a particle moving freely on [[3-sphere|the surface of a four-dimensional (hyper-)sphere]]<!--a 3-manifold, embedded in 4-space; the latter may be clearer to our readers-->,<ref name="fock_1935" >{{cite journal | last = Fock | first = V. | author-link = Vladimir Fock | date = 1935 | title = Zur Theorie des Wasserstoffatoms | journal = Zeitschrift für Physik | volume = 98 | issue = 3–4 | pages = 145–154 | doi = 10.1007/BF01336904|bibcode = 1935ZPhy...98..145F | s2cid = 123112334 }}</ref> so that the whole problem is symmetric under certain rotations of the four-dimensional space.<ref name="bargmann_1936" >{{cite journal | last = Bargmann | first = V. | author-link = Valentine Bargmann | date = 1936 | title = Zur Theorie des Wasserstoffatoms: Bemerkungen zur gleichnamigen Arbeit von V. Fock | journal = Zeitschrift für Physik | volume = 99 | issue = 7–8 | pages = 576–582 | doi = 10.1007/BF01338811 | bibcode = 1936ZPhy...99..576B | s2cid = 117461194 }}</ref> 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.<ref name="hamilton_1847_hodograph">{{cite journal | last = Hamilton | first = W. R. | author-link = William Rowan Hamilton | date = 1847 | title = The hodograph or a new method of expressing in symbolic language the Newtonian law of attraction | journal = Proceedings of the Royal Irish Academy | volume = 3 | pages = 344–353 }}</ref> The Laplace–Runge–Lenz vector is named after [[Pierre-Simon Laplace|Pierre-Simon de Laplace]], [[Carl David Tolmé Runge|Carl Runge]] and [[Wilhelm Lenz]]. It is also known as the '''Laplace vector''',<ref name="goldstein_1980c">{{cite book | last=Goldstein | first=H. | author-link=Herbert Goldstein | date=1980 | title=Classical Mechanics | edition=2nd | publisher=Addison Wesley | page=421}}</ref><ref name="arnold_1989b">{{cite book | last = Arnold | first = V. I. | author-link = Vladimir Arnold | date = 1989 | title = Mathematical Methods of Classical Mechanics | edition = 2nd | publisher = Springer-Verlag | location = New York | pages = 413–415 | isbn = 0-387-96890-3 }}</ref> the '''Runge–Lenz vector'''<ref name="goldstein_1975_1976">{{cite journal | last=Goldstein | first=H. | author-link=Herbert Goldstein | date=1975 | title=Prehistory of the Runge–Lenz vector | journal=[[American Journal of Physics]] | volume=43 | issue=8 | pages=737–738 | doi=10.1119/1.9745|bibcode = 1975AmJPh..43..737G }}<br />{{cite journal | last=Goldstein | first=H. | author-link=Herbert Goldstein | date=1976 | title=More on the prehistory of the Runge–Lenz vector | journal=[[American Journal of Physics]] | volume=44 | issue=11 | pages=1123–1124 | doi=10.1119/1.10202|bibcode = 1976AmJPh..44.1123G }}</ref> and the '''Lenz vector'''.<ref name="bohm_1993" /> Ironically, [[Stigler's law of eponymy|none of those scientists]] discovered it.<ref name="goldstein_1975_1976" /> The LRL vector has been re-discovered and re-formulated several times;<ref name="goldstein_1975_1976" /> for example, it is equivalent to the dimensionless [[eccentricity vector]] of [[celestial mechanics]].<ref name="taff_1985" /><ref name="arnold_1989b" /><ref name="hamilton_1847_quaternions">{{cite journal | last = Hamilton | first = W. R. | author-link = William Rowan Hamilton | date = 1847 | title = Applications of Quaternions to Some Dynamical Questions | journal = Proceedings of the Royal Irish Academy | volume = 3 | pages = Appendix III}}</ref> Various generalizations of the LRL vector have been defined, which incorporate the effects of [[special relativity]], [[electromagnetic field]]s and even different types of central forces.<ref name="landau_lifshitz_1976">{{cite book | last=Landau | first=L. D. | author-link=Lev Landau | author2=Lifshitz E. M. | author-link2=Evgeny Lifshitz | date=1976 | title=Mechanics | edition=3rd | publisher=Pergamon Press | page=[https://archive.org/details/mechanics00land/page/154 154] | isbn=0-08-021022-8 | url=https://archive.org/details/mechanics00land/page/154 }}</ref><ref name="fradkin_1967">{{cite journal | last = Fradkin | first = D. M. | date = 1967 | title = Existence of the Dynamic Symmetries O<sub>4</sub> and SU<sub>3</sub> for All Classical Central Potential Problems | journal = Progress of Theoretical Physics | volume = 37 | issue = 5 | pages = 798–812 | doi = 10.1143/PTP.37.798|bibcode = 1967PThPh..37..798F | doi-access = free }}</ref><ref name="yoshida_1987">{{cite journal | last = Yoshida | first = T. | date = 1987 | title = Two methods of generalisation of the Laplace–Runge–Lenz vector | journal = European Journal of Physics | volume = 8 | issue = 4 | pages = 258–259 | doi = 10.1088/0143-0807/8/4/005|bibcode = 1987EJPh....8..258Y | s2cid = 250843588 }}</ref>
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