Editing Laplace–Runge–Lenz vector

{{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>

==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.

== 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" />

== Definition ==
An inverse-square central force acting on a single particle is described by the equation 
<math display="block">
\mathbf{F}(r)=-\frac{k}{r^{2}}\mathbf{\hat{r}};
</math>
The corresponding [[potential energy]] is given by <math>V(r) = - k / r</math>. The constant parameter {{mvar|k}} describes the strength of the central force; it is equal to {{math|[[Gravitational constant|''G'']]''Mm''}} for gravitational and {{math|−{{sfrac|1|4''π''[[Vacuum electric permittivity|''ε''<sub>0</sub>]]}}''Qq''}} for electrostatic forces. The force is attractive if {{math|''k'' > 0}} and repulsive if {{math|''k'' < 0}}.

[[File:Laplace Runge Lenz vector.svg|thumb|right|400px|Figure 1: The LRL vector {{math|'''A'''}} (shown in red) at four points (labeled 1, 2, 3 and 4) on the elliptical orbit of a bound point particle moving under an inverse-square central force. The center of attraction is shown as a small black circle from which the position vectors (likewise black) emanate. The angular momentum vector {{math|'''L'''}} is perpendicular to the orbit. The coplanar vectors {{math|'''p''' × '''L'''}} and {{math|(''mk''/''r'')'''r'''}} are shown in blue and green, respectively; these variables are defined [[#Mathematical definition|below]]. The vector {{math|'''A'''}} is constant in direction and magnitude.]]

The LRL vector {{math|'''A'''}} is defined mathematically by the formula<ref name="goldstein_1980" />
{{Equation box 1
|indent =:
|equation = <math> \mathbf{A} = \mathbf{p} \times \mathbf{L} - m k \mathbf{\hat{r}},</math>
|cellpadding= 6
|border
|border colour = #0073CF
|background colour=#F9FFF7}}
where
* {{mvar|m}} is the [[mass]] of the point particle moving under the central force,
* {{math|'''p'''}} is its momentum vector,
* {{math|1='''L''' = '''r''' × '''p'''}} is its angular momentum vector,
* {{math|'''r'''}} is the position vector of the particle (Figure 1),
* <math>\mathbf{\hat{r}}</math> is the corresponding [[unit vector]], i.e., <math>\mathbf{\hat{r}} = \frac{\mathbf{r}}{r}</math>, and
* {{mvar|r}} is the magnitude of {{math|'''r'''}}, the distance of the mass from the center of force.

The [[International System of Units|SI units]] of the LRL vector are joule-kilogram-meter (J⋅kg⋅m). This follows because the units of {{math|'''p'''}} and {{math|'''L'''}} are kg⋅m/s and J⋅s, respectively. This agrees with the units of {{mvar|m}} (kg) and of {{mvar|k}} (N⋅m<sup>2</sup>).

This definition of the LRL vector {{math|'''A'''}} pertains to a single point particle of mass {{mvar|m}} moving under the action of a fixed force. However, the same definition may be extended to [[two-body problem]]s such as the Kepler problem, by taking {{mvar|m}} as the [[reduced mass]] of the two bodies and {{math|'''r'''}} as the vector between the two bodies.

Since the assumed force is conservative, the total energy {{mvar|E}} is a constant of motion,
<math display="block">
E = \frac{p^{2}}{2m} - \frac{k}{r} = \frac{1}{2} mv^{2} - \frac{k}{r}.
</math>

The assumed force is also a central force. Hence, the angular momentum vector {{math|'''L'''}} is also conserved and defines the plane in which the particle travels. The LRL vector {{math|'''A'''}} is perpendicular to the angular momentum vector {{math|'''L'''}} because both {{math|'''p''' × '''L'''}} and {{math|'''r'''}} are perpendicular to {{math|'''L'''}}. It follows that {{math|'''A'''}} lies in the plane of motion.

Alternative formulations for the same constant of motion may be defined, typically by scaling the vector with constants, such as the mass {{math|''m''}}, the force parameter {{math|''k''}} or the angular momentum {{math|''L''}}.<ref name="goldstein_1975_1976" /> The most common variant is to divide {{math|'''A'''}} by {{math|''mk''}}, which yields the eccentricity vector,<ref name="taff_1985" /><ref name="hamilton_1847_quaternions" /> a [[dimensionless quantity|dimensionless]] vector along the semi-major axis whose modulus equals the eccentricity of the conic:
<math display="block">
\mathbf{e} = \frac{\mathbf{A}}{m k} = \frac{1}{m k}(\mathbf{p} \times \mathbf{L}) - \mathbf{\hat{r}}.
</math>
An equivalent formulation<ref name="arnold_1989b" /> multiplies this eccentricity vector by the major semiaxis {{mvar|a}}, giving the resulting vector the units of length. Yet another formulation<ref name="symon_1971b">{{cite book | last=Symon | first=K. R. | author-link=Keith Symon | date=1971 | title=Mechanics | edition=3rd | publisher=Addison Wesley | pages=130–131}}</ref> divides {{math|'''A'''}} by <math>L^2</math>, yielding an equivalent conserved quantity with units of inverse length, a quantity that appears in the solution of the Kepler problem
<math display="block">
u \equiv \frac{1}{r} = \frac{km}{L^2} + \frac{A}{L^2} \cos\theta
</math>
where <math>\theta</math> is the angle between {{math|'''A'''}} and the position vector {{math|'''r'''}}. Further alternative formulations are given [[#Alternative scalings, symbols and formulations|below]].

== Derivation of the Kepler orbits ==
[[File:Laplace Runge Lenz vector2.svg|thumb|right|350px|Figure 2: Simplified version of Figure 1, defining the angle {{mvar|θ}} between {{math|'''A'''}} and {{math|'''r'''}} at one point of the orbit.]]

The ''shape'' and ''orientation'' of the orbits can be determined from the LRL vector as follows.<ref name="goldstein_1980" /> Taking the dot product of {{math|'''A'''}} with the position vector {{math|'''r'''}} gives the equation
<math display="block">
\mathbf{A} \cdot \mathbf{r} = A \cdot r \cdot \cos\theta = 
\mathbf{r} \cdot \left( \mathbf{p} \times \mathbf{L} \right) - mkr,
</math>
where {{mvar|θ}} is the angle between {{math|'''r'''}} and {{math|'''A'''}} (Figure 2). Permuting the [[triple product|scalar triple product]] yields
<math display="block">
\mathbf{r} \cdot\left(\mathbf{p}\times \mathbf{L}\right) = 
\left(\mathbf{r} \times \mathbf{p}\right)\cdot\mathbf{L} = 
\mathbf{L}\cdot\mathbf{L}=L^2
</math>

Rearranging yields the solution for the Kepler equation
{{Equation box 1
|indent =:
|equation = <math>
\frac{1}{r} = \frac{mk}{L^2} + \frac{A}{L^{2}} \cos\theta</math>
|cellpadding= 6
|border
|border colour = #0073CF
|background colour=#F9FFF7}}

This corresponds to the formula for a conic section of [[eccentricity (orbit)|eccentricity]] ''e''
<math display="block">
\frac{1}{r} = C \cdot \left( 1 + e \cdot \cos\theta \right)
</math>
where the eccentricity <math>e = \frac{A}{\left| mk \right|} \geq 0</math> and {{mvar|C}} is a constant.<ref name="goldstein_1980" />

Taking the dot product of {{math|'''A'''}} with itself yields an equation involving the total energy {{mvar|E}},<ref name="goldstein_1980" />
<math display="block">
A^2 = m^2 k^2 + 2 m E L^2,
</math>
which may be rewritten in terms of the eccentricity,<ref name="goldstein_1980" />
<math display="block">
e^{2} = 1 + \frac{2L^2}{mk^2}E.
</math>

Thus, if the energy {{mvar|E}} is negative (bound orbits), the eccentricity is less than one and the orbit is an ellipse. Conversely, if the energy is positive (unbound orbits, also called "scattered orbits"<ref name="goldstein_1980" />), the eccentricity is greater than one and the orbit is a [[hyperbola]].<ref name="goldstein_1980" /> Finally, if the energy is exactly zero, the eccentricity is one and the orbit is a [[parabola]].<ref name="goldstein_1980" /> In all cases, the direction of {{math|'''A'''}} lies along the symmetry axis of the conic section and points from the center of force toward the periapsis, the point of closest approach.<ref name="goldstein_1980" />

== Circular momentum hodographs ==
[[File:Kepler hodograph3.svg|thumb|right|280px|Figure 3: The momentum vector {{math|'''p'''}} (shown in blue) moves on a circle as the particle moves on an ellipse. The four labeled points correspond to those in Figure 1. The circle is centered on the {{mvar|y}}-axis at position {{math|''A''/''L''}} (shown in magenta), with radius {{math|''mk''/''L''}} (shown in green). The angle η determines the eccentricity {{mvar|e}} of the elliptical orbit ({{math|1=cos ''η'' = ''e''}}). By the [[inscribed angle|inscribed angle theorem]] for [[circle]]s, {{mvar|η}} is also the angle between any point on the circle and the two points of intersection with the {{math|''p''<sub>''x''</sub>}} axis, {{math|1=''p''<sub>''x''</sub> = ±''p''<sub>0</sub>}}, which only depend on {{mvar|E}}, but not {{mvar|L}}.]]

The conservation of the LRL vector {{math|'''A'''}} and angular momentum vector {{math|'''L'''}} is useful in showing that the momentum vector {{math|'''p'''}} moves on a [[circle]] under an inverse-square central force.<ref name="hamilton_1847_hodograph" /><ref name="goldstein_1975_1976" />

Taking the dot product of
<math display="block"> mk \hat{\mathbf{r}} = \mathbf{p} \times \mathbf{L} - \mathbf{A} </math>
with itself yields
<math display="block"> (mk)^2= A^2+ p^2 L^2 + 2 \mathbf{L} \cdot (\mathbf{p} \times \mathbf{A}). </math>

Further choosing {{math|'''L'''}} along the {{mvar|z}}-axis, and the major semiaxis as the {{mvar|x}}-axis, yields the locus equation for {{math|'''p'''}},
{{Equation box 1
|indent =:
|equation = <math> p_x^2 + \left(p_y - \frac A L \right)^2 = \left( \frac{mk} L \right)^2.</math>
|cellpadding= 6
|border
|border colour = #0073CF
|background colour=#F9FFF7}}

In other words, the momentum vector {{math|'''p'''}} is confined to a circle of radius {{math|1=''mk''/''L'' = ''L''/''ℓ''}} centered on {{math|(0, ''A''/''L'')}}.<ref>The conserved binormal Hamilton vector <math>\mathbf{B}\equiv \mathbf{L} \times \mathbf{A} / L^2</math> on this momentum plane (pink) has a simpler geometrical significance, and may actually supplant it, as <math>\mathbf{A} =\mathbf {B}\times \mathbf {L}</math>, see Patera, R. P. (1981). "Momentum-space derivation of the Runge-Lenz vector", ''Am. J. Phys'' '''49''' 593–594. It has length {{math|''A''/''L''}} and is discussed in section [[#Alternative scalings, symbols and formulations]].</ref> For bounded orbits, the eccentricity {{mvar|e}} corresponds to the cosine of the angle {{mvar|η}} shown in Figure 3.  For unbounded orbits, we have <math> A > m k</math> and so the circle does not intersect the <math>p_x</math>-axis.

In the degenerate limit of circular orbits, and thus vanishing {{math|'''A'''}}, the circle centers at the origin {{math|(0,0)}}.
For brevity, it is also useful to introduce the variable <math display="inline">p_0 = \sqrt{2m|E|}</math>.

This circular [[hodograph]] is useful in illustrating the symmetry of the Kepler problem.

== Constants of motion and superintegrability ==
The seven scalar quantities {{mvar|E}}, {{math|'''A'''}} and {{math|'''L'''}} (being vectors, the latter two contribute three conserved quantities each) are related by two equations, {{math|1='''A''' ⋅ '''L''' = 0}} and {{math|1=''A''<sup>2</sup> = ''m''<sup>2</sup>''k''<sup>2</sup> + 2 ''mEL''<sup>2</sup>}}, giving five independent [[constants of motion]]. (Since the magnitude of {{math|'''A'''}}, hence the eccentricity {{mvar|e}} of the orbit, can be determined from the total angular momentum {{mvar|L}} and the energy {{mvar|E}}, only the ''direction'' of {{math|'''A'''}} is conserved independently; moreover, since {{math|'''A'''}} must be perpendicular to {{math|'''L'''}}, it contributes ''only one'' additional conserved quantity.)

This is consistent with the six initial conditions (the particle's initial position and velocity vectors, each with three components) that specify the orbit of the particle, since the initial time is not determined by a constant of motion. The resulting 1-dimensional orbit in 6-dimensional phase space is thus completely specified.

A mechanical system with {{mvar|d}} degrees of freedom can have at most {{math|2''d'' − 1}} constants of motion, since there are {{math|2''d''}} initial conditions and the initial time cannot be determined by a constant of motion. A system with more than {{mvar|d}} constants of motion is called ''superintegrable'' and a system with {{math|2''d'' − 1}} constants is called '''[[Superintegrable Hamiltonian system|maximally superintegrable]]'''.<ref>{{cite journal | last = Evans | first = N. W. | date = 1990 | title = Superintegrability in classical mechanics | journal = Physical Review A | volume = 41 | issue = 10 | pages = 5666–5676 | doi = 10.1103/PhysRevA.41.5666| pmid = 9902953 |bibcode = 1990PhRvA..41.5666E }}</ref> Since the solution of the [[Hamilton–Jacobi equation]] in one [[coordinate system]] can yield only {{mvar|d}} constants of motion, superintegrable systems must be separable in more than one coordinate system.<ref>{{cite book | last = Sommerfeld | first = A. | author-link = Arnold Sommerfeld | date = 1923 | title = Atomic Structure and Spectral Lines | publisher = Methuen | location = London | page = 118}}</ref>{{irrelevant citation|date=May 2025|reason=That chapter in Sommerfeld does not discuss separability at all, nor have I been able to find such a discussion elsewhere in the book.}} The Kepler problem is maximally superintegrable, since it has three degrees of freedom ({{math|1=''d'' = 3}}) and five independent constant of motion; its Hamilton–Jacobi equation is separable in both [[spherical coordinates]] and [[parabolic coordinates]],<ref name="landau_lifshitz_1976" /> as described [[#Hamilton–Jacobi equation in parabolic coordinates|below]].

Maximally superintegrable systems follow closed, one-dimensional orbits in [[phase space]], since the orbit is the intersection of the phase-space [[isosurface]]s of their constants of motion. Consequently, the orbits are perpendicular to all gradients of all these independent isosurfaces, five in this specific problem, and hence are determined by the generalized cross products of all of these gradients. As a result, '''all''' superintegrable systems are automatically describable by [[Nambu mechanics]],<ref>{{cite journal | last = Curtright | first= T. |author2=Zachos C. | date= 2003 | title=Classical and Quantum Nambu Mechanics | journal = Physical Review | volume = D68 | issue= 8 | page = 085001 | doi = 10.1103/PhysRevD.68.085001|arxiv = hep-th/0212267 |bibcode = 2003PhRvD..68h5001C | s2cid= 17388447 }}</ref> alternatively, and equivalently, to [[Hamiltonian mechanics]].

Maximally superintegrable systems can be [[canonical quantization|quantized]] using [[commutation relation]]s, as illustrated [[#Quantum mechanics of the hydrogen atom|below]].<ref>{{cite journal | last = Evans | first = N. W. | date = 1991 | title = Group theory of the Smorodinsky–Winternitz system | journal = Journal of Mathematical Physics | volume = 32 | issue = 12 | pages = 3369–3375 | doi = 10.1063/1.529449 | bibcode = 1991JMP....32.3369E }}</ref> Nevertheless, equivalently, they are also quantized in the Nambu framework, such as this classical Kepler problem into the quantum hydrogen atom.<ref>{{cite journal | title= Branes, quantum Nambu brackets, and the hydrogen atom | last = Zachos | first = C. | author2=Curtright T. | date= 2004 | journal = Czech Journal of Physics | volume= 54 | issue = 11 | pages = 1393–1398 | doi= 10.1007/s10582-004-9807-x |arxiv = math-ph/0408012 | bibcode = 2004CzJPh..54.1393Z | s2cid = 14074249 }}</ref>

== Evolution under perturbed potentials ==
[[File:Relativistic precession.svg|right|thumb|250px|Figure 5: Gradually precessing elliptical orbit, with an eccentricity ''e''&nbsp;=&nbsp;0.667. Such precession arises in the Kepler problem if the attractive central force deviates slightly from an inverse-square law. The ''rate'' of precession can be calculated using the formulae in the text.]]

The Laplace–Runge–Lenz vector {{math|'''A'''}} is conserved only for a perfect inverse-square central force. In most practical problems such as planetary motion, however, the interaction potential energy between two bodies is not exactly an inverse square law, but may include an additional central force, a so-called ''perturbation'' described by a potential energy {{math|''h''(''r'')}}. In such cases, the LRL vector rotates slowly in the plane of the orbit, corresponding to a slow [[apsidal precession]] of the orbit.

By assumption, the perturbing potential {{math|''h''(''r'')}} is a conservative central force, which implies that the total energy {{mvar|E}} and angular momentum vector {{math|'''L'''}} are conserved. Thus, the motion still lies in a plane perpendicular to {{math|'''L'''}} and the magnitude {{mvar|A}} is conserved, from the equation {{math|1=''A''<sup>2</sup> = ''m''<sup>2</sup>''k''<sup>2</sup> + 2''mEL''<sup>2</sup>}}. The perturbation potential {{math|''h''(''r'')}} may be any sort of function, but should be significantly weaker than the main inverse-square force between the two bodies.

The ''rate'' at which the LRL vector rotates provides information about the perturbing potential {{math|''h''(''r'')}}. Using canonical perturbation theory and [[action-angle coordinates]], it is straightforward to show<ref name="goldstein_1980" /> that {{math|'''A'''}} rotates at a rate of,
<math display="block">\begin{align}
\frac{\partial}{\partial L} \langle h(r) \rangle & = \frac{\partial}{\partial L} \left\{ \frac{1}{T} \int_0^T h(r) \, dt \right\} \\[1em]
& = \frac{\partial}{\partial L} \left\{ \frac{m}{L^{2}} \int_0^{2\pi} r^2 h(r) \, d\theta \right\},
\end{align}</math>
where {{mvar|T}} is the orbital period, and the identity {{math|1=''L'' ''dt'' = ''m'' ''r''<sup>2</sup> ''dθ''}} was used to convert the time integral into an angular integral (Figure 5). The expression in angular brackets, {{math|{{langle}}''h''(''r''){{rangle}}}}, represents the perturbing potential, but ''averaged'' over one full period; that is, averaged over one full passage of the body around its orbit. Mathematically, this time average corresponds to the following quantity in curly braces. This averaging helps to suppress fluctuations in the rate of rotation.

This approach was used to help verify [[Albert Einstein|Einstein's]] theory of [[general relativity]], which adds a small effective inverse-cubic perturbation to the normal Newtonian gravitational potential,<ref name="einstein_1915">{{cite journal | last = Einstein | first = A. | author-link = Albert Einstein | date = 1915 | title = Erklärung der Perihelbewegung des Merkur aus der allgemeinen Relativitätstheorie | journal = Sitzungsberichte der Preussischen Akademie der Wissenschaften | volume = 1915 | pages = 831–839| bibcode = 1915SPAW.......831E }}</ref>
<math display="block">
h(r) = \frac{kL^{2}}{m^{2}c^{2}} \left( \frac{1}{r^{3}} \right).
</math>

Inserting this function into the integral and using the equation
<math display="block">
\frac{1}{r} = \frac{mk}{L^2} \left( 1 + \frac{A}{mk} \cos\theta \right)
</math>
to express {{mvar|r}} in terms of {{mvar|θ}}, the precession rate of the periapsis caused by this non-Newtonian perturbation is calculated to be<ref name="einstein_1915" />
<math display="block">
\frac{6 \pi k^2}{T L^2 c^2},
</math>
which closely matches the observed anomalous precession of [[Mercury (planet)|Mercury]]<ref>{{cite journal | last = Le Verrier | first = U. J. J. | author-link = Urbain Le Verrier | date = 1859 | title = Lettre de M. Le Verrier à M. Faye sur la Théorie de Mercure et sur le Mouvement du Périhélie de cette Planète | journal = Comptes Rendus de l'Académie des Sciences de Paris | volume = 49 | pages = 379–383}}</ref> and binary [[pulsar]]s.<ref>{{cite book | last = Will | first = C. M. | date = 1979 | title = General Relativity, an Einstein Century Survey | edition = SW Hawking and W Israel | publisher = Cambridge University Press | location = Cambridge | pages = Chapter 2 | no-pp = true}}</ref> This agreement with experiment is strong evidence for general relativity.<ref>{{cite book | last = Pais | first = A. | author-link = Abraham Pais | date = 1982 | title = Subtle is the Lord: The Science and the Life of Albert Einstein | url = https://archive.org/details/subtleislordscie00pais | url-access = registration | publisher = Oxford University Press }}</ref><ref>{{cite book | last = Roseveare | first = N. T. | date = 1982 | title = Mercury's Perihelion from Le Verrier to Einstein | url = https://archive.org/details/mercurysperiheli0000rose | url-access = registration | publisher = Oxford University Press| isbn = 978-0-19-858174-1 }}</ref>

== Poisson brackets ==
=== Unscaled functions ===
The algebraic structure of the problem is, as explained in later sections, {{math|1=SO(4)/'''Z'''<sub>2</sub> ~ SO(3) × SO(3)}}.<ref name="bargmann_1936" />
The three components ''L<sub>i</sub>'' of the angular momentum vector {{math|'''L'''}} have the Poisson brackets<ref name="goldstein_1980" />
<math display="block">
\{ L_i, L_j\} = \sum_{s=1}^3 \varepsilon_{ijs} L_s,
</math>
where {{mvar|i}}=1,2,3 and {{math|''ε<sub>ijs</sub>''}} is the fully [[antisymmetric tensor]], i.e., the [[Levi-Civita symbol]]; the summation index {{mvar|s}} is used here to avoid confusion with the force parameter {{mvar|k}} defined [[#Mathematical definition|above]]. Then since the LRL vector {{math|'''A'''}} transforms like a vector, we have the following Poisson bracket relations between {{math|'''A'''}} and {{math|'''L'''}}:<ref>{{harvnb|Hall|2013}} Proposition 17.25.</ref>
<math display="block">\{A_i,L_j\}=\sum_{s=1}^3\varepsilon_{ijs}A_s.</math> 
Finally, the Poisson bracket relations between the different components of {{math|'''A'''}} are as follows:<ref>{{harvnb|Hall|2013}} Proposition 18.7; note that Hall uses a different normalization of the LRL vector.</ref>
<math display="block">\{A_i,A_j\}=-2mH\sum_{s=1}^3\varepsilon_{ijs}L_s,</math>
where <math>H</math> is the Hamiltonian. Note that the span of the components of {{math|'''A'''}} and the components of {{math|'''L'''}} is not closed under Poisson brackets, because of the factor of <math>H</math> on the right-hand side of this last relation.

Finally, since both {{math|'''L'''}} and {{math|'''A'''}} are constants of motion, we have
<math display="block">\{A_i, H\} = \{L_i, H\} = 0.</math>

The Poisson brackets will be extended to quantum mechanical [[canonical commutation relation|commutation relations]] in the [[#Quantum mechanics of the hydrogen atom|next section]] and to [[Lie algebra|Lie bracket]]s in a [[#Conservation and symmetry|following section]].

=== Scaled functions ===
As noted [[#Alternative scalings, symbols and formulations|below]], a scaled Laplace–Runge–Lenz vector {{math|'''D'''}} may be defined with the same units as angular momentum by dividing {{math|'''A'''}} by <math display="inline">p_0 = \sqrt{2m|H|}</math>. Since {{math|'''D'''}} still transforms like a vector, the Poisson brackets of {{math|'''D'''}} with the angular momentum vector {{math|'''L'''}} can then be written in a similar form<ref name="bargmann_1936" /><ref name="bohm_1993" />
<math display="block">
\{ D_i, L_j\} = \sum_{s=1}^3 \varepsilon_{ijs} D_s.
</math>

The Poisson brackets of {{math|'''D'''}} with ''itself'' depend on the [[sign (mathematics)|sign]] of {{mvar|H}}, i.e., on whether the energy is [[negative number|negative]] (producing closed, elliptical orbits under an inverse-square central force) or [[positive number|positive]] (producing open, hyperbolic orbits under an inverse-square central force). For ''negative'' energies—i.e., for bound systems—the Poisson brackets are<ref name="Hall 2013">{{harvnb|Hall|2013}} Theorem 18.9.</ref>
<math display="block">
\{ D_i, D_j\} = \sum_{s=1}^3 \varepsilon_{ijs} L_s.
</math>
We may now appreciate the motivation for the chosen scaling of {{math|'''D'''}}: With this scaling, the Hamiltonian no longer appears on the right-hand side of the preceding relation. Thus, the span of the three components of {{math|'''L'''}} and the three components of {{math|'''D'''}} forms a six-dimensional Lie algebra under the Poisson bracket. This Lie algebra is isomorphic to {{math|so(4)}}, the Lie algebra of the 4-dimensional rotation group {{math|SO(4)}}.<ref name="ReferenceA">{{harvnb|Hall|2013|at=Section 18.4.4.}}</ref>

By contrast, for ''positive'' energy, the Poisson brackets have the opposite sign,
<math display="block">
\{ D_i, D_j\} = -\sum_{s=1}^3 \varepsilon_{ijs} L_s.
</math>
In this case, the Lie algebra is isomorphic to {{math|so(3,1)}}.

The distinction between positive and negative energies arises because the desired scaling—the one that eliminates the Hamiltonian from the right-hand side of the Poisson bracket relations between the components of the scaled LRL vector—involves the ''square root'' of the Hamiltonian. To obtain real-valued functions, we must then take the absolute value of the Hamiltonian, which distinguishes between positive values (where <math>|H| = H</math>) and negative values (where <math>|H| = -H</math>).

=== Laplace-Runge-Lenz operator for the hydrogen atom in momentum space ===
Scaled Laplace-Runge-Lenz operator in the momentum space was found in 2022 .<ref>{{cite journal |last1=Efimov |first1=S.P. |title=Coordinate space modification of Fock's theory. Harmonic tensors in the quantum Coulomb problem |journal=Physics-Uspekhi |date=2022 |volume=65 |issue=9 |pages=952–967 |doi=10.3367/UFNe.2021.04.038966|bibcode=2022PhyU...65..952E |s2cid=234871720 |arxiv=2501.00010 }}</ref><ref>{{cite journal |last1=Efimov |first1=S.P. |title=Runge-Lenz Operator in the Momentum Space |journal=JETP Letters |date=2023 |volume=117 |issue=9 |pages=716–720 |doi=10.1134/S0021364023600635|bibcode=2023JETPL.117..716E |s2cid=259225778 |doi-access=free |arxiv=2411.14482 }}</ref> The formula for the operator is simpler than in position space:
: <math > \hat \mathbf{A}_{\mathbf p}=\imath(\hat l_{\mathbf p}+1 )\mathbf p -\frac{(p^2+1)}{2}\imath\mathbf\nabla_{\mathbf p  } ,   </math>
where the "degree operator" 
: <math > \hat l_{\mathbf p }=(\mathbf p \mathbf \nabla_{\mathbf p} )  </math>
multiplies a homogeneous polynomial by its degree.

=== Casimir invariants and the energy levels ===
The [[Casimir invariant]]s for negative energies are
<math display="block"> \begin{align}
C_1 &= \mathbf{D} \cdot \mathbf{D} + \mathbf{L} \cdot \mathbf{L} = \frac{mk^2}{2|E|}, \\
C_2 &= \mathbf{D} \cdot \mathbf{L} = 0,
\end{align}</math>
and have vanishing Poisson brackets with all components of {{math|'''D'''}} and {{math|'''L'''}},
<math display="block">
\{ C_1, L_i \} = \{ C_1, D_i\} = 
\{ C_2, L_i \} = \{ C_2, D_i \} = 0.
</math>
''C''<sub>2</sub> is trivially zero, since the two vectors are always perpendicular.

However, the other invariant, ''C''<sub>1</sub>, is non-trivial and depends only on {{mvar|m}}, {{mvar|k}} and {{mvar|E}}. Upon canonical quantization, this invariant allows the energy levels of [[hydrogen-like atom]]s to be derived using only quantum mechanical canonical commutation relations, instead of the conventional solution of the Schrödinger equation.<ref name="bohm_1993" /><ref name="ReferenceA" /> This derivation is discussed in detail in the next section.

== Quantum mechanics of the hydrogen atom ==
<!-- This section is linked from [[Dipole]] -->

[[File:Hydrogen energy levels.png|thumb|right|300px|Figure 6: Energy levels of the hydrogen atom as predicted from the commutation relations of angular momentum and Laplace–Runge–Lenz vector operators; these energy levels have been verified experimentally.]]

Poisson brackets provide a simple guide for quantizing most classical systems: the commutation relation of two quantum mechanical [[operator (physics)|operator]]s is specified by the Poisson bracket of the corresponding classical variables, multiplied by {{math|''iħ''}}.<ref>{{cite book | last = Dirac | first = P. A. M. | author-link = Paul Dirac | date = 1958 | title = Principles of Quantum Mechanics | edition = 4th revised | publisher = Oxford University Press}}</ref>

By carrying out this quantization and calculating the eigenvalues of the {{mvar|C}}<sub>1</sub> [[Casimir element|Casimir operator]] for the Kepler problem, Wolfgang Pauli was able to derive the [[energy spectrum|energy levels]] of hydrogen-like atoms (Figure 6) and, thus, their atomic emission spectrum.<ref name="pauli_1926" /> This elegant 1926 derivation was obtained ''before the development of the Schrödinger equation''.<ref>{{cite journal | last = Schrödinger | first = E. | author-link = Erwin Schrödinger | date = 1926 | title = Quantisierung als Eigenwertproblem | journal = Annalen der Physik | volume = 384 | issue = 4 | pages = 361–376 | doi = 10.1002/andp.19263840404 |bibcode = 1926AnP...384..361S | doi-access = free }}</ref>

A subtlety of the quantum mechanical operator for the LRL vector {{math|'''A'''}} is that the momentum and angular momentum operators do not commute; hence, the quantum operator cross product of {{math|'''p'''}} and {{math|'''L'''}} must be defined carefully.<ref name="bohm_1993" /> Typically, the operators for the Cartesian components {{math|''A<sub>s</sub>''}} are defined using a symmetrized (Hermitian) product,
<math display="block">
A_s = - m k \hat{r}_s + \frac{1}{2} \sum_{i=1}^3 \sum_{j=1}^3 \varepsilon_{sij} (p_i \ell_j + \ell_j p_i),
</math>
Once this is done, one can show that the quantum LRL operators satisfy commutations relations exactly analogous to the Poisson bracket relations in the previous section—just replacing the Poisson bracket with <math>1/(i\hbar)</math> times the commutator.<ref>{{harvnb|Hall|2013}} Proposition 18.12.</ref><ref>{{Cite book |last=Merzbacher|first=Eugen | url= https://books.google.com/books?id=6Ja_QgAACAAJ | title=Quantum Mechanics | date=1998-01-07 | publisher=John Wiley & Sons | isbn=978-0-471-88702-7 | pages=268–270 | language=en |author-link=Eugen Merzbacher}}</ref>

From these operators, additional [[ladder operators]] for {{math|'''L'''}} can be defined,
<math display="block">\begin{align}
J_0 &= A_3, \\
J_{\pm 1} &= \mp \tfrac{1}{\sqrt{2}} \left( A_1 \pm i A_2 \right).
\end{align}</math>
These further connect ''different'' eigenstates of {{math|'''L'''<sup>2</sup>}}, so different spin multiplets, among themselves.

A normalized first Casimir invariant operator, quantum analog of the above, can likewise be defined,
<math display="block">C_1 = - \frac{m k^2}{2 \hbar^{2}} H^{-1} - I,</math>
where {{math|''H''<sup>−1</sup>}} is the inverse of the [[Hamiltonian (quantum mechanics)|Hamiltonian]] energy operator, and {{mvar|I}} is the [[identity function|identity operator]].

Applying these ladder operators to the [[eigenstate]]s |''ℓ''{{math|''mn''}}〉 of the total angular momentum, azimuthal angular momentum and energy operators, the eigenvalues of the first Casimir operator, {{mvar|C}}<sub>1</sub>, are seen to be quantized, {{math|''n''<sup>2</sup> − 1}}. Importantly, by dint of the vanishing of ''C''<sub>2</sub>, they are independent of the ℓ and {{mvar|m}} quantum numbers, making the [[degenerate energy level|energy levels degenerate]].<ref name="bohm_1993" />

Hence, the energy levels are given by
<math display="block">
E_n = - \frac{m k^2}{2\hbar^{2} n^2},
</math>
which coincides with the [[Rydberg formula]] for hydrogen-like atoms (Figure 6). The additional symmetry operators {{math|'''A'''}} have connected the different ℓ multiplets among themselves, for a given energy (and ''C''<sub>1</sub>), dictating {{math|''n''<sup>2</sup>}} states at each level. In effect, they have enlarged the angular momentum group {{math|[[SO(3)]]}} to {{math|[[SO(4)]]/'''Z'''<sub>2</sub> ~ SO(3) × SO(3)}}.<ref>{{harvnb|Hall|2013}} Theorem 18.14.</ref>

== Conservation and symmetry ==
The conservation of the LRL vector corresponds to a subtle symmetry of the system. In [[classical mechanics]], symmetries are continuous operations that map one orbit onto another without changing the energy of the system; in quantum mechanics, symmetries are continuous operations that "mix" [[atomic orbital|electronic orbitals]] of the same energy, i.e., degenerate energy levels. A conserved quantity is usually associated with such symmetries.<ref name="goldstein_1980" /> For example, every central force is symmetric under the [[rotation group SO(3)]], leading to the conservation of the angular momentum {{math|'''L'''}}. Classically, an overall rotation of the system does not affect the energy of an orbit; quantum mechanically, rotations mix the [[spherical harmonic]]s of the same [[quantum number]] {{mvar|ℓ}} without changing the energy.

[[File:Kepler hodograph family.png|thumb|right|Figure 7: The family of circular momentum hodographs for a given energy {{mvar|E}}. All the circles pass through the same two points <math display="inline">\pm p_0 = \pm \sqrt{2m|E|}</math> on the ''p''<sub>''x''</sub> axis (see Figure 3). This family of hodographs corresponds to one family of [[Apollonian circles]], and the σ isosurfaces of [[bipolar coordinates]].]]

The symmetry for the inverse-square central force is higher and more subtle. The peculiar symmetry of the Kepler problem results in the conservation of both the angular momentum vector {{math|'''L'''}} and the LRL vector {{math|'''A'''}} (as defined [[#Mathematical definition|above]]) and, quantum mechanically, ensures that the energy levels of hydrogen do not depend on the angular momentum quantum numbers {{mvar|ℓ}} and {{mvar|m}}. The symmetry is more subtle, however, because the symmetry operation must take place in a [[dimension|higher-dimensional space]]; such symmetries are often called "hidden symmetries".<ref name="prince_eliezer_1981" />

Classically, the higher symmetry of the Kepler problem allows for continuous alterations of the orbits that preserve energy but not angular momentum; expressed another way, orbits of the same energy but different angular momentum (eccentricity) can be transformed continuously into one another. Quantum mechanically, this corresponds to mixing orbitals that differ in the {{mvar|ℓ}} and {{mvar|m}} quantum numbers, such as the {{mvar|s}}({{math|1=''ℓ'' = 0}}) and {{mvar|p}}({{math|1=''ℓ'' = 1}}) atomic orbitals. Such mixing cannot be done with ordinary three-dimensional translations or rotations, but is equivalent to a rotation in a higher dimension.

For ''negative'' energies – i.e., for bound systems – the higher symmetry group is {{math|[[SO(4)]]}}, which preserves the length of four-dimensional vectors
<math display="block">
|\mathbf{e}|^2 = e_1^2 + e_2^2 + e_3^2 + e_4^2.
</math>

In 1935, [[Vladimir Fock]] showed that the quantum mechanical bound Kepler problem is equivalent to the problem of a free particle confined to a [[3-sphere|three-dimensional unit sphere]] in four-dimensional space.<ref name="fock_1935" /> Specifically, Fock showed that the Schrödinger [[wavefunction]] in the momentum space for the Kepler problem was the [[stereographic projection]] of the spherical harmonics on the sphere. Rotation of the sphere and re-projection results in a continuous mapping of the elliptical orbits without changing the energy, an {{math|SO(4)}} symmetry sometimes known as Fock symmetry;<ref>{{cite journal |last1=Nikitin |first1=A G |title=New exactly solvable systems with Fock symmetry |journal=Journal of Physics A: Mathematical and Theoretical |date=7 December 2012 |volume=45 |issue=48 |pages=485204 |doi=10.1088/1751-8113/45/48/485204|arxiv=1205.3094 |bibcode=2012JPhA...45V5204N |s2cid=119138270 }}</ref> quantum mechanically, this corresponds to a mixing of all orbitals of the same energy quantum number {{mvar|n}}. [[Valentine Bargmann]] noted subsequently that the Poisson brackets for the angular momentum vector {{math|'''L'''}} and the scaled LRL vector {{math|'''A'''}} formed the Lie algebra for {{math|SO(4)}}.<ref name="bargmann_1936" /><ref name="Hall 2013"/> Simply put, the six quantities {{math|'''A'''}} and {{math|'''L'''}} correspond to the six conserved angular momenta in four dimensions, associated with the six possible simple rotations in that space (there are six ways of choosing two axes from four). This conclusion does not imply that our [[universe]] is a three-dimensional sphere; it merely means that this particular physics problem (the two-body problem for inverse-square central forces) is ''mathematically equivalent'' to a free particle on a three-dimensional sphere.

For ''positive'' energies –&nbsp;i.e., for unbound, "scattered" systems&nbsp;– the higher symmetry group is {{math|[[SO(3,1)]]}}, which preserves the [[Minkowski space|Minkowski length]] of [[4-vector]]s
<math display="block">
ds^2 = e_1^2 + e_2^2 + e_3^2 - e_4^2.
</math>

Both the negative- and positive-energy cases were considered by Fock<ref name="fock_1935" /> and Bargmann<ref name="bargmann_1936" /> and have been reviewed encyclopedically by Bander and Itzykson.<ref name="bander_itzykson_1966">{{cite journal | last = Bander | first = M. |author2=Itzykson C. | date = 1966 | title = Group Theory and the Hydrogen Atom (I) | journal = Reviews of Modern Physics | volume = 38 | issue = 2 | pages = 330–345 | doi = 10.1103/RevModPhys.38.330 | bibcode=1966RvMP...38..330B| url = https://escholarship.org/uc/item/42q1m0t7 }}</ref><ref>{{cite journal | last = Bander | first = M. |author2=Itzykson C. | date = 1966 | title = Group Theory and the Hydrogen Atom (II) | journal = Reviews of Modern Physics | volume = 38 | issue = 2 | pages = 346–358 | doi = 10.1103/RevModPhys.38.346 | bibcode=1966RvMP...38..346B| url = https://escholarship.org/uc/item/4ns2m34s }}</ref>

The orbits of central-force systems&nbsp;– and those of the Kepler problem in particular&nbsp;– are also symmetric under [[reflection (mathematics)|reflection]]. Therefore, the {{math|SO(3)}}, {{math|SO(4)}} and {{math|SO(3,1)}} groups cited above are not the full symmetry groups of their orbits; the full groups are [[orthogonal group|{{math|O(3)}}]], {{math|O(4)}}, and [[O(3,1)]], respectively. Nevertheless, only the [[connected space|connected]] [[subgroup]]s, {{math|SO(3)}}, {{math|SO(4)}}, and {{math|SO<sup>+</sup>(3,1)}}, are needed to demonstrate the conservation of the angular momentum and LRL vectors; the reflection symmetry is irrelevant for conservation, which may be derived from the Lie algebra of the group.

== Rotational symmetry in four dimensions ==
[[File:Kepler Fock projection.svg|thumb|right|300px|Figure 8: The momentum hodographs of Figure 7 correspond to stereographic projections of [[great circle]]s on the three-dimensional {{mvar|η}} unit sphere. All of the great circles intersect the {{math|''η<sub>x</sub>''}} axis, which is perpendicular to the page; the projection is from the North pole (the {{math|'''w'''}} unit vector) to the {{math|''η''<sub>''x''</sub>}}−{{math|''η''<sub>''y''</sub>}} plane, as shown here for the magenta hodograph by the dashed black lines. The great circle at a latitude {{mvar|α}} corresponds to an [[eccentricity (mathematics)|eccentricity]] {{math|1=''e'' = sin ''α''}}. The colors of the great circles shown here correspond to their matching hodographs in Figure 7.]]

The connection between the Kepler problem and four-dimensional rotational symmetry {{math|SO(4)}} can be readily visualized.<ref name="bander_itzykson_1966" /><ref name="rogers_1973">{{cite journal | last = Rogers | first = H. H. | date = 1973 | title = Symmetry transformations of the classical Kepler problem | journal = Journal of Mathematical Physics | volume = 14 | issue = 8 | pages = 1125–1129 | doi = 10.1063/1.1666448|bibcode = 1973JMP....14.1125R }}</ref><ref>{{cite book | last = Guillemin | first = V. |author2=Sternberg S. | date = 1990 | title = Variations on a Theme by Kepler | publisher = American Mathematical Society Colloquium Publications | volume = 42 | isbn = 0-8218-1042-1}}</ref> Let the four-dimensional Cartesian coordinates be denoted {{math|(''w'', ''x'', ''y'', ''z'')}} where {{math|(''x'', ''y'', ''z'')}} represent the Cartesian coordinates of the normal position vector {{math|'''r'''}}. The three-dimensional momentum vector {{math|'''p'''}} is associated with a four-dimensional vector <math>\boldsymbol\eta</math> on a three-dimensional unit sphere
<math display="block">\begin{align}
\boldsymbol\eta & = \frac{p^2 - p_0^2}{p^2 + p_0^2} \mathbf{\hat{w}} + \frac{2 p_0}{p^2 + p_0^2} \mathbf{p} \\[1em]
& = \frac{mk - r p_0^2}{mk} \mathbf{\hat{w}} + \frac{rp_0}{mk} \mathbf{p},
\end{align}</math>
where <math>\mathbf{\hat{w}}</math> is the unit vector along the new {{mvar|w}} axis. The transformation mapping {{math|'''p'''}} to {{math|'''η'''}} can be uniquely inverted; for example, the {{mvar|x}} component of the momentum equals
<math display="block"> p_x = p_0 \frac{\eta_x}{1 - \eta_w}, </math>
and similarly for {{math|''p<sub>y</sub>''}} and {{math|''p<sub>z</sub>''}}. In other words, the three-dimensional vector {{math|'''p'''}} is a stereographic projection of the four-dimensional <math>\boldsymbol\eta</math> vector, scaled by {{math|''p''<sub>0</sub>}} (Figure 8).

Without loss of generality, we may eliminate the normal rotational symmetry by choosing the Cartesian coordinates such that the {{mvar|z}} axis is aligned with the angular momentum vector {{math|'''L'''}} and the momentum hodographs are aligned as they are in Figure 7, with the centers of the circles on the {{mvar|y}} axis. Since the motion is planar, and {{math|'''p'''}} and {{math|'''L'''}} are perpendicular, {{math|1=''p<sub>z</sub>'' = ''η''<sub>''z''</sub> = 0}} and attention may be restricted to the three-dimensional vector {{nowrap|1=<math>\boldsymbol\eta = (\eta_w, \eta_x, \eta_y)</math>.}} The family of [[Apollonian circles]] of momentum hodographs (Figure 7) correspond to a family of [[great circle]]s on the three-dimensional <math>\boldsymbol\eta</math> sphere, all of which intersect the {{math|''η''<sub>''x''</sub>}} axis at the two foci {{math|1=''η''<sub>''x''</sub> = ±1}}, corresponding to the momentum hodograph foci at {{math|1=''p<sub>x</sub>'' = ±''p''<sub>0</sub>}}. These great circles are related by a simple rotation about the {{math|''η''<sub>''x''</sub>}}-axis (Figure 8). This rotational symmetry transforms all the orbits of the same energy into one another; however, such a rotation is orthogonal to the usual three-dimensional rotations, since it transforms the fourth dimension {{math|''η''<sub>''w''</sub>}}. This higher symmetry is characteristic of the Kepler problem and corresponds to the conservation of the LRL vector.

An elegant [[action-angle variables]] solution for the Kepler problem can be obtained by eliminating the redundant four-dimensional coordinates <math>\boldsymbol\eta</math> in favor of elliptic cylindrical coordinates {{math|(''χ'', ''ψ'', ''φ'')}}<ref>{{cite journal | last = Lakshmanan | first = M. |author2=Hasegawa H. | title = On the canonical equivalence of the Kepler problem in coordinate and momentum spaces | journal = Journal of Physics A | volume = 17 | issue = 16 | pages = L889–L893 | doi = 10.1088/0305-4470/17/16/006 | date = 1984|bibcode = 1984JPhA...17L.889L }}</ref>
<math display="block">\begin{align}
\eta_w &= \operatorname{cn} \chi \operatorname{cn} \psi, \\[1ex]
\eta_x &= \operatorname{sn} \chi \operatorname{dn} \psi \cos \phi, \\[1ex]
\eta_y &= \operatorname{sn} \chi \operatorname{dn} \psi \sin \phi, \\[1ex]
\eta_z &= \operatorname{dn} \chi \operatorname{sn} \psi,
\end{align}</math>
where {{math|sn}}, {{math|cn}} and {{math|dn}} are [[Jacobi's elliptic functions]].

== Generalizations to other potentials and relativity ==
The Laplace–Runge–Lenz vector can also be generalized to identify conserved quantities that apply to other situations.

In the presence of a uniform electric field {{math|'''E'''}}, the generalized Laplace–Runge–Lenz vector <math>\mathcal{A}</math> is<ref name="landau_lifshitz_1976" /><ref>{{cite journal | last = Redmond | first = P. J. | date = 1964 | title = Generalization of the Runge–Lenz Vector in the Presence of an Electric Field | journal = Physical Review | volume = 133 | issue = 5B | pages = B1352–B1353 | doi = 10.1103/PhysRev.133.B1352|bibcode = 1964PhRv..133.1352R }}</ref>
<math display="block">
\mathcal{A} = \mathbf{A} + \frac{mq}{2} \left[ \left( \mathbf{r} \times \mathbf{E} \right) \times \mathbf{r} \right],
</math>
where {{mvar|q}} is the [[electric charge|charge]] of the orbiting particle. Although <math>\mathcal{A}</math> is not conserved, it gives rise to a conserved quantity, namely <math>\mathcal{A} \cdot \mathbf{E}</math>.

Further generalizing the Laplace–Runge–Lenz vector to other potentials and [[special relativity]], the most general form can be written as<ref name="fradkin_1967" />
<math display="block">
\mathcal{A} = 
\left( \frac{\partial \xi}{\partial u} \right) \left(\mathbf{p} \times \mathbf{L}\right) +
\left[ \xi - u \left( \frac{\partial \xi}{\partial u} \right)\right] L^{2} \mathbf{\hat{r}},
</math>
where {{math|1=''u'' = 1/''r''}} and {{math|1=''ξ'' = cos ''θ''}}, with the angle {{mvar|θ}} defined by
<math display="block">
\theta = L \int^u \frac{du}{\sqrt{m^2 c^2 (\gamma^2 - 1) - L^2 u^{2}}},
</math>
and {{mvar|γ}} is the [[Lorentz factor]]. As before, we may obtain a conserved binormal vector {{math|'''B'''}} by taking the cross product with the conserved angular momentum vector
<math display="block">
\mathcal{B} = \mathbf{L} \times \mathcal{A}.
</math>

These two vectors may likewise be combined into a conserved [[dyadic tensor]] {{math|'''W'''}},
<math display="block">
\mathcal{W} = \alpha \mathcal{A} \otimes \mathcal{A} + \beta \, \mathcal{B} \otimes \mathcal{B}.
</math>

In illustration, the LRL vector for a non-relativistic, isotropic harmonic oscillator can be calculated.<ref name="fradkin_1967" /> Since the force is central,
<math display="block">
\mathbf{F}(r)= -k \mathbf{r},
</math>
the angular momentum vector is conserved and the motion lies in a plane.

The conserved dyadic tensor can be written in a simple form
<math display="block">
\mathcal{W} = \frac{1}{2m} \mathbf{p} \otimes \mathbf{p} + \frac{k}{2} \, \mathbf{r} \otimes \mathbf{r},
</math>
although {{math|'''p'''}} and {{math|'''r'''}} are not necessarily perpendicular.

The corresponding Runge–Lenz vector is more complicated,
<math display="block">
\mathcal{A} = \frac{1}{\sqrt{mr^2 \omega_0 A - mr^2 E + L^2}} \left\{ \left( \mathbf{p} \times \mathbf{L} \right) + \left(mr\omega_0 A - mrE \right) \mathbf{\hat{r}} \right\},
</math>
where 
<math display="block">\omega_0 = \sqrt{\frac{k}{m}}</math> 
is the natural oscillation frequency, and 
<math display="block">A = (E^2-\omega^2 L^2)^{1/2} / \omega.</math>

== Proofs that the Laplace–Runge–Lenz vector is conserved in Kepler problems ==
The following are arguments showing that the LRL vector is conserved under central forces that obey an inverse-square law.

=== Direct proof of conservation ===
A central force <math>\mathbf{F}</math> acting on the particle is
<math display="block">
\mathbf{F} = \frac{d\mathbf{p}}{dt} = f(r) \frac{\mathbf{r}}{r} = f(r) \mathbf{\hat{r}}
</math>
for some function <math>f(r)</math> of the radius <math>r</math>. Since the angular momentum <math>\mathbf{L} = \mathbf{r} \times \mathbf{p}</math> is conserved under central forces, <math display="inline">\frac{d}{dt}\mathbf{L} = 0</math> and
<math display="block">
\frac{d}{dt} \left( \mathbf{p} \times \mathbf{L} \right) = \frac{d\mathbf{p}}{dt} \times \mathbf{L} = f(r) \mathbf{\hat{r}} \times \left( \mathbf{r} \times m \frac{d\mathbf{r}}{dt} \right) = f(r) \frac{m}{r} \left[ \mathbf{r} \left(\mathbf{r} \cdot \frac{d\mathbf{r}}{dt} \right) - r^2 \frac{d\mathbf{r}}{dt} \right],
</math>
where the momentum <math display="inline">\mathbf{p} = m \frac{d\mathbf{r}}{dt}</math> and where the triple cross product has been simplified using [[Vector triple product|Lagrange's formula]]
<math display="block">
\mathbf{r} \times \left( \mathbf{r} \times \frac{d\mathbf{r}}{dt} \right) = \mathbf{r} \left(\mathbf{r} \cdot \frac{d\mathbf{r}}{dt} \right) - r^2 \frac{d\mathbf{r}}{dt}.
</math>

The identity
<math display="block">
\frac{d}{dt} \left( \mathbf{r} \cdot \mathbf{r} \right) = 2 \mathbf{r} \cdot \frac{d\mathbf{r}}{dt} = \frac{d}{dt} (r^2) = 2r\frac{dr}{dt}
</math>
yields the equation
<math display="block">
\frac{d}{dt} \left( \mathbf{p} \times \mathbf{L} \right) = 
-m f(r) r^2 \left[ \frac{1}{r} \frac{d\mathbf{r}}{dt} - \frac{\mathbf{r}}{r^2} \frac{dr}{dt}\right] = -m f(r) r^2 \frac{d}{dt} \left( \frac{\mathbf{r}}{r}\right).
</math>

For the special case of an inverse-square central force <math display="inline">f(r)=\frac{-k}{r^{2}}</math>, this equals
<math display="block">
\frac{d}{dt} \left( \mathbf{p} \times \mathbf{L} \right) = 
m k \frac{d}{dt} \left( \frac{\mathbf{r}}{r}\right) = 
\frac{d}{dt} (mk\mathbf{\hat{r}}).
</math>

Therefore, {{math|'''A'''}} is conserved for inverse-square central forces<ref>{{harvnb|Hall|2013}} Proposition 2.34.</ref>
<math display="block">
\frac{d}{dt} \mathbf{A} = \frac{d}{dt} \left( \mathbf{p} \times \mathbf{L} \right) - \frac{d}{dt} \left( mk\mathbf{\hat{r}} \right) = \mathbf{0}.
</math>

A shorter proof is obtained by using the relation of angular momentum to angular velocity, <math> \mathbf{L} = m r^2 \boldsymbol{\omega}</math>, which holds for a particle traveling in a plane perpendicular to <math> \mathbf{L}</math>. Specifying to inverse-square central forces, the time derivative of <math>\mathbf{p} \times \mathbf{L}</math> is
<math display="block">
 \frac{d}{dt} \mathbf{p} \times \mathbf{L} = \left( \frac{-k}{r^2} \mathbf{\hat{r}} \right) \times \left(m r^2 \boldsymbol{\omega}\right)
= m k \, \boldsymbol{\omega} \times \mathbf{\hat{r}} = m k \,\frac{d}{dt}\mathbf{\hat{r}},
</math>
where the last equality holds because a unit vector can only change by rotation, and <math>\boldsymbol{\omega}\times\mathbf{\hat{r}}</math> is the orbital velocity of the rotating vector. Thus, {{math|'''A'''}} is seen to be a difference of two vectors with equal time derivatives.

As described [[#Generalizations to other potentials and relativity|elsewhere in this article]], this LRL vector {{math|'''A'''}} is a special case of a general conserved vector <math>\mathcal{A}</math> that can be defined for all central forces.<ref name="fradkin_1967" /><ref name="yoshida_1987" /> However, since most central forces do not produce closed orbits (see [[Bertrand's theorem]]), the analogous vector <math>\mathcal{A}</math> rarely has a simple definition and is generally a [[multivalued function]] of the angle {{mvar|θ}} between {{math|'''r'''}} and <math>\mathcal{A}</math>.

=== Hamilton–Jacobi equation in parabolic coordinates ===
The constancy of the LRL vector can also be derived from the Hamilton–Jacobi equation in parabolic coordinates {{math|1=(''ξ'', ''η'')}}, which are defined by the equations
<math display="block">\begin{align}
\xi &= r + x, \\
\eta &= r - x,
\end{align}</math>
where {{mvar|r}} represents the radius in the plane of the orbit
<math display="block">r = \sqrt{x^2 + y^2}.</math>

The inversion of these coordinates is
<math display="block">\begin{align}
x &= \tfrac{1}{2} (\xi - \eta), \\
y &= \sqrt{\xi\eta},
\end{align}</math>

Separation of the Hamilton–Jacobi equation in these coordinates yields the two equivalent equations<ref name="landau_lifshitz_1976" /><ref>{{cite journal | last = Dulock | first = V. A. |author2=McIntosh H. V. |author2-link=Harold V. McIntosh| date = 1966 | title = On the Degeneracy of the Kepler Problem | journal = Pacific Journal of Mathematics | volume = 19 | pages = 39–55 | doi=10.2140/pjm.1966.19.39| doi-access = free }}</ref>

<math display="block"> \begin{align}
2\xi p_\xi^2 - mk - mE\xi &= -\Gamma, \\
2\eta p_\eta^2 - mk - mE\eta &= \Gamma,
\end{align}</math>
where {{math|Γ}} is a constant of motion. Subtraction and re-expression in terms of the Cartesian momenta {{math|''p''<sub>''x''</sub>}} and {{math|''p''<sub>''y''</sub>}} shows that {{math|Γ}} is equivalent to the LRL vector
<math display="block">
\Gamma = p_y (x p_y - y p_x) - mk\frac{x}{r} = A_x.
</math>

=== Noether's theorem ===
The connection between the rotational symmetry described above and the conservation of the LRL vector can be made quantitative by way of [[Noether's theorem]]. This theorem, which is used for finding constants of motion, states that any infinitesimal variation of the [[generalized coordinate]]s of a physical system
<math display="block">
\delta q_i = \varepsilon g_i(\mathbf{q}, \mathbf{\dot{q}}, t)
</math>
that causes the Lagrangian to vary to first order by a total time derivative
<math display="block">
\delta L = \varepsilon \frac{d}{dt} G(\mathbf{q}, t)
</math>
corresponds to a conserved quantity {{math|Γ}}
<math display="block">
\Gamma = -G + \sum_i g_i \left( \frac{\partial L}{\partial \dot{q}_i}\right).
</math>

In particular, the conserved LRL vector component {{math|''A<sub>s</sub>''}} corresponds to the variation in the coordinates<ref>{{cite journal | last = Lévy-Leblond | first = J. M. | date = 1971 | title = Conservation Laws for Gauge-Invariant Lagrangians in Classical Mechanics | journal = American Journal of Physics | volume = 39 | issue = 5 | pages = 502–506 | doi = 10.1119/1.1986202 | bibcode = 1971AmJPh..39..502L }}</ref>
<math display="block">
\delta_s x_i = \frac{\varepsilon}{2} \left[ 2 p_i x_s - x_i p_s - \delta_{is} \left( \mathbf{r} \cdot \mathbf{p} \right) \right],
</math>
where {{mvar|i}} equals 1, 2 and 3, with {{math|''x<sub>i</sub>''}} and {{math|''p<sub>i</sub>''}} being the {{mvar|i}}-th components of the position and momentum vectors {{math|'''r'''}} and {{math|'''p'''}}, respectively; as usual, {{math|''δ<sub>is</sub>''}} represents the [[Kronecker delta]]. The resulting first-order change in the Lagrangian is
<math display="block">
\delta L = \frac{1}{2}\varepsilon mk\frac{d}{dt} \left( \frac{x_s}{r} \right).
</math>

Substitution into the general formula for the conserved quantity {{math|Γ}} yields the conserved component {{math|''A<sub>s</sub>''}} of the LRL vector,
<math display="block">
A_s = \left[ p^2 x_s - p_s \ \left(\mathbf{r} \cdot \mathbf{p}\right) \right] - mk \left( \frac{x_s}{r} \right) = 
\left[ \mathbf{p} \times \left( \mathbf{r} \times \mathbf{p} \right) \right]_s - mk \left( \frac{x_s}{r} \right).
</math>

=== Lie transformation ===
[[File:Scaled ellipses.png|thumb|right|350px|Figure 9: The Lie transformation from which the conservation of the LRL vector {{math|'''A'''}} is derived. As the scaling parameter {{mvar|λ}} varies, the energy and angular momentum changes, but the eccentricity {{mvar|e}} and the magnitude and direction of {{math|'''A'''}} do not.]]

[[Noether's theorem]] derivation of the conservation of the LRL vector {{math|'''A'''}} is elegant, but has one drawback: the coordinate variation {{math|''δx''<sub>''i''</sub>}} involves not only the ''position'' {{math|'''r'''}}, but also the ''momentum'' {{math|'''p'''}} or, equivalently, the ''velocity'' {{math|'''v'''}}.<ref>{{cite journal | last = Gonzalez-Gascon | first = F. | date = 1977 | title = Notes on the symmetries of systems of differential equations | journal = Journal of Mathematical Physics | volume = 18 | issue = 9 | pages = 1763–1767 | doi = 10.1063/1.523486|bibcode = 1977JMP....18.1763G }}</ref> This drawback may be eliminated by instead deriving the conservation of {{math|'''A'''}} using an approach pioneered by [[Sophus Lie]].<ref>{{cite book | last = Lie | first = S. | author-link = Sophus Lie | date = 1891 | title = Vorlesungen über Differentialgleichungen | url = https://archive.org/details/vorlesuberdiff00liesrich | publisher = Teubner | location = Leipzig}}</ref><ref>{{cite book | last = Ince | first = E. L. | date = 1926 | title = Ordinary Differential Equations | publisher = Dover (1956 reprint) | location = New York | pages = 93–113}}</ref> Specifically, one may define a Lie transformation<ref name="prince_eliezer_1981" >{{cite journal | last = Prince | first = G. E. |author2=Eliezer C. J. | date = 1981 | title = On the Lie symmetries of the classical Kepler problem | journal = Journal of Physics A: Mathematical and General | volume = 14 | issue = 3 | pages = 587–596 | doi = 10.1088/0305-4470/14/3/009|bibcode = 1981JPhA...14..587P }}</ref> in which the coordinates {{math|'''r'''}} and the time {{mvar|t}} are scaled by different powers of a parameter λ (Figure 9),
<math display="block">
t \rightarrow \lambda^{3}t , \qquad \mathbf{r} \rightarrow \lambda^{2}\mathbf{r} , \qquad\mathbf{p} \rightarrow \frac{1}{\lambda}\mathbf{p}.
</math>

This transformation changes the total angular momentum {{mvar|L}} and energy {{mvar|E}},
<math display="block">
L \rightarrow \lambda L, \qquad E \rightarrow \frac{1}{\lambda^{2}} E,
</math>
but preserves their product ''EL''<sup>2</sup>. Therefore, the eccentricity {{mvar|e}} and the magnitude {{mvar|A}} are preserved, as may be seen from the [[#Derivation of the Kepler orbits|equation for {{math|''A''<sup>2</sup>}}]]
<math display="block">A^2 = m^2 k^2 e^{2} = m^2 k^2 + 2 m E L^2.</math>

The direction of {{math|'''A'''}} is preserved as well, since the semiaxes are not altered by a global scaling. This transformation also preserves [[Kepler's third law]], namely, that the semiaxis {{mvar|a}} and the period {{mvar|T}} form a constant&nbsp;{{math|''T''<sup>2</sup>/''a''<sup>3</sup>}}.

== Alternative scalings, symbols and formulations ==
Unlike the momentum and angular momentum vectors {{math|'''p'''}} and {{math|'''L'''}}, there is no universally accepted definition of the Laplace–Runge–Lenz vector; several different scaling factors and symbols are used in the scientific literature. The most common definition is given [[#Mathematical definition|above]], but another common alternative is to divide by the quantity {{math|''mk''}} to obtain a dimensionless conserved eccentricity vector
<math display="block">
\mathbf{e} = 
\frac{1}{mk} \left(\mathbf{p} \times \mathbf{L} \right) - \mathbf{\hat{r}} = 
\frac{m}{k} \left(\mathbf{v} \times \left( \mathbf{r} \times \mathbf{v} \right) \right) - \mathbf{\hat{r}},
</math>
where {{math|'''v'''}} is the velocity vector. This scaled vector {{math|'''e'''}} has the same direction as {{math|'''A'''}} and its magnitude equals the eccentricity of the orbit, and thus vanishes for circular orbits.

Other scaled versions are also possible, e.g., by dividing {{math|'''A'''}} by {{mvar|m}} alone
<math display="block">
\mathbf{M} = \mathbf{v} \times \mathbf{L} - k\mathbf{\hat{r}},
</math>
or by {{math|''p''<sub>0</sub>}}
<math display="block">
\mathbf{D} = \frac{\mathbf{A}}{p_{0}} = 
\frac{1}{\sqrt{2m|E|}}
\left( \mathbf{p} \times \mathbf{L} - m k \mathbf{\hat{r}} \right),
</math>
which has the same units as the angular momentum vector {{math|'''L'''}}.

In rare cases, the sign of the LRL vector may be reversed, i.e., scaled by {{math|−1}}. Other common symbols for the LRL vector include {{math|'''a'''}}, {{math|'''R'''}}, {{math|'''F'''}}, {{math|'''J'''}} and {{math|'''V'''}}. However, the choice of scaling and symbol for the LRL vector do not affect its conservation.

[[File:Kepler trivector.svg|thumb|right|250px|Figure 4: The angular momentum vector {{math|'''L'''}}, the LRL vector {{math|'''A'''}} and Hamilton's vector, the binormal {{math|'''B'''}}, are mutually perpendicular; {{math|'''A'''}} and {{math|'''B'''}} point along the major and minor axes, respectively, of an elliptical orbit of the Kepler problem.]]

An alternative conserved vector is the [[binormal]] vector {{math|'''B'''}} studied by William Rowan Hamilton,<ref name="hamilton_1847_quaternions" />
{{Equation box 1
|indent =:
|equation = <math>
\mathbf{B} = \mathbf{p} - \frac{mk}{L^2 r} \left( \mathbf{L} \times \mathbf{r} \right),
</math>
|cellpadding= 6
|border
|border colour = #0070BF
|bgcolor=#FAFFFB}}
which is conserved and points along the ''minor'' semiaxis of the ellipse. (It is not defined for vanishing eccentricity.)

The LRL vector {{math|1='''A''' = '''B''' × '''L'''}} is the cross product of {{math|'''B'''}} and {{math|'''L'''}} (Figure 4). On the momentum hodograph in the relevant section above, {{math|'''B'''}} is readily seen to connect the origin of momenta with the center of the circular hodograph, and to possess magnitude {{math|''A''/''L''}}. At perihelion, it points in the direction of the momentum.

The vector {{math|'''B'''}} is denoted as "binormal" since it is perpendicular to both {{math|'''A'''}} and {{math|'''L'''}}. Similar to the LRL vector itself, the binormal vector can be defined with different scalings and symbols.

The two conserved vectors, {{math|'''A'''}} and {{math|'''B'''}} can be combined to form a conserved dyadic tensor {{math|'''W'''}},<ref name="fradkin_1967" />
<math display="block">
\mathbf{W} = \alpha \mathbf{A} \otimes \mathbf{A} + \beta \, \mathbf{B} \otimes \mathbf{B},
</math>
where {{mvar|α}} and {{mvar|β}} are arbitrary scaling constants and <math>\otimes</math> represents the [[tensor product]] (which is not related to the [[cross product|vector cross product]], despite their similar symbol). Written in explicit components, this equation reads
<math display="block">
W_{ij} = \alpha A_i A_j + \beta B_i B_j.
</math>

Being perpendicular to each another, the vectors {{math|'''A'''}} and {{math|'''B'''}} can be viewed as the [[principal axis (mechanics)|principal axes]] of the conserved [[tensor]] {{math|'''W'''}}, i.e., its scaled [[eigenvector]]s. {{math|'''W'''}} is perpendicular to {{math|'''L'''}} ,
<math display="block">
\mathbf{L} \cdot \mathbf{W} = 
\alpha \left( \mathbf{L} \cdot \mathbf{A} \right) \mathbf{A} + \beta \left( \mathbf{L} \cdot \mathbf{B} \right) \mathbf{B} = 0,
</math>
since {{math|'''A'''}} and {{math|'''B'''}} are both perpendicular to {{math|'''L'''}} as well, {{math|1='''L''' ⋅ '''A''' = '''L''' ⋅ '''B''' = 0}}.

More directly, this equation reads, in explicit components,
<math display="block">
\left( \mathbf{L} \cdot \mathbf{W} \right)_j = \alpha \left( \sum_{i=1}^3 L_i A_i \right) A_j + \beta \left( \sum_{i=1}^3 L_i B_i \right) B_j = 0.
</math>

== See also ==
* [[Astrodynamics]]
** [[Orbit]]
** [[Eccentricity vector]]
** [[Orbital elements]]
* [[Bertrand's theorem]]
* [[Binet equation]]
* [[Two-body problem]]

== References ==
{{reflist|30em}}

== Further reading ==
* {{cite web | first=John | last=Baez | author-link=John Baez | date = 2008 | url=https://math.ucr.edu/home/baez/classical/runge.pdf | title=The Kepler Problem Revisited: The Laplace–Runge–Lenz Vector | access-date=2021-05-31}}
* {{cite web | first=John | last=Baez | author-link=John Baez | date = 2003 | url=http://math.ucr.edu/home/baez/gravitational.html | title=Mysteries of the gravitational 2-body problem | access-date=2004-12-11 | archive-url=https://web.archive.org/web/20081021082335/http://math.ucr.edu/home/baez/gravitational.html | archive-date=2008-10-21 | url-status=dead }}
* {{cite web | first=John | last=Baez | author-link=John Baez | date = 2018 | url=http://math.ucr.edu/home/baez/gravitational.html | title=Mysteries of the gravitational 2-body problem | access-date=2021-05-31}} Updated version of previous source.
* {{cite journal| last = D'Eliseo| first = M. M.| date = 2007| title = The first-order orbital equation| journal = American Journal of Physics| volume = 75| issue = 4| pages = 352–355| doi = 10.1119/1.2432126|bibcode = 2007AmJPh..75..352D }}
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[[Category:Classical mechanics]]
[[Category:Orbits]]
[[Category:Rotational symmetry]]
[[Category:Vectors (mathematics and physics)]]
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[[Category:Mathematical physics]]