Laplace–Runge–Lenz vector

Laplace–Runge–Lenz vector

: "Throughout this article, vectors and their magnitudes are indicated by boldface and italic type, respectively; for example, left| mathbf{A} ight| = A."

In classical mechanics, the Laplace–Runge–Lenz vector (or simply the LRL vector) is a vector used chiefly to describe the shape and orientation of the orbit of one astronomical body around another, such as a planet revolving around a sun. For two bodies interacting by Newtonian gravity, the LRL vector is a constant of motion, meaning that it is the same no matter where it is calculated on the orbit;cite book | last=Goldstein | first=H. | authorlink=Herbert Goldstein | year=1980 | title=Classical Mechanics | edition=2nd edition | publisher=Addison Wesley | pages=102–105,421–422] equivalently, the LRL vector is said to be "conserved". More generally, the LRL vector is conserved in all problems in which two bodies interact by a central force that varies as the inverse square of the distance between them; such problems are called Kepler problems. [cite book | last = Arnold | first = VI | authorlink = Vladimir Arnold | year = 1989 | title = Mathematical Methods of Classical Mechanics, 2nd ed. | publisher = Springer-Verlag | location = New York | pages = 38 | id = ISBN 0-387-96890-3]

The hydrogen atom is a Kepler problem, since it comprises two charged particles interacting by Coulomb's law of electrostatics, another inverse square central force. The LRL vector was essential in the first quantum mechanical derivation of the spectrum of the hydrogen atom,cite journal | last = Pauli | first = W | authorlink = Wolfgang Pauli | year = 1926 | title = Über das Wasserstoffspektrum vom Standpunkt der neuen Quantenmechanik | journal = Zeitschrift für Physik | volume = 36 | pages = 336–363 | doi = 10.1007/BF01450175] before the development of the Schrödinger equation. However, this approach is rarely used today.

In classical and quantum mechanics, conserved quantities generally correspond to a symmetry of the system. The conservation of the LRL vector corresponds to an unusual symmetry; the Kepler problem is mathematically equivalent to a particle moving freely on the boundary of a four-dimensional ball ,cite journal | last = Fock | first = V | authorlink = Vladimir Fock | year = 1935 | title = Zur Theorie des Wasserstoffatoms | journal = Zeitschrift für Physik | volume = 98 | pages = 145–154 | doi = 10.1007/BF01336904] so that the whole problem is symmetric under certain rotations of the four-dimensional space.cite journal | last = Bargmann | first = V | authorlink = Valentine Bargmann | year = 1936 | title = Zur Theorie des Wasserstoffatoms: Bemerkungen zur gleichnamigen Arbeit von V. Fock | journal = Zeitschrift für Physik | volume = 99 | pages = 576–582 | doi = 10.1007/BF01338811] 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 another in the same two points.cite journal | last = Hamilton | first = WR | authorlink = William Rowan Hamilton | year = 1847 | title = Unknown title | journal = Proceedings of the Royal Irish Academy | volume = 3 | pages = 344ff]

The Laplace–Runge–Lenz vector is named after Pierre-Simon de Laplace, Carle Runge and Wilhelm Lenz. It is also known as the Laplace vector, the Runge–Lenz vector and the Lenz vector. Ironically, none of those scientists discovered it. The LRL vector has been re-discovered several timescite journal | last=Goldstein | first=H. | authorlink=Herbert Goldstein | year=1975 | title=Prehistory of the Runge–Lenz vector | journal=American Journal of Physics | volume=43 | pages=735–738 | doi=10.1119/1.9745
cite journal | last=Goldstein | first=H. | authorlink=Herbert Goldstein | year=1976 | title=More on the prehistory of the Runge–Lenz vector | journal=American Journal of Physics | volume=44 | pages=1123–1124 | doi=10.1119/1.10202] and is also equivalent to the dimensionless eccentricity vector of celestial mechanics.cite journal | last = Hamilton | first = WR | authorlink = William Rowan Hamilton | year = 1847 | title = Applications of Quaternions to Some Dynamical Questions | journal = Proceedings of the Royal Irish Academy | volume = 3 | pages = Appendix III] Various generalizations of the LRL vector have been defined, which incorporate the effects of special relativity, electromagnetic fields and even different types of central forces.


A single particle moving under any conservative central force has at least four constants of motion, the total energy "E" and the three Cartesian components of the angular momentum vector L. The particle's orbit is confined to a plane defined by the particle's initial momentum p (or, equivalently, its velocity v) and the vector r between the particle and the center of force (see Figure 1, below).

As defined below (see Mathematical definition), the Laplace–Runge–Lenz vector (LRL vector) A always lies in the plane of motion for any central force. However, A is constant only for an inverse-square central force. For most central forces, however, this vector A is not constant, but changes in both length and direction; if the central force is "approximately" an inverse-square law, the vector A is approximately constant in length, but slowly rotates its direction. A "generalized" conserved LRL vector mathcal{A} can be defined for all central forces, but this generalized vector is a complicated function of position, and usually not expressible in closed form.cite journal | last = Fradkin | first = DM | year = 1967 | title = Existence of the Dynamic Symmetries O4 and SU3 for All Classical Central Potential Problems | journal = Progress of Theoretical Physics | volume = 37 | pages = 798–812 | doi = 10.1143/PTP.37.798] cite journal | last = Yoshida | first = T | year = 1987 | title = Two methods of generalisation of the Laplace–Runge–Lenz vector | journal = European Journal of Physics | volume = 8 | pages = 258–259 | doi = 10.1088/0143-0807/8/4/005]

The plane of motion is perpendicular to the angular momentum vector L, which is constant; this may be expressed mathematically by the vector dot product equation r·L = 0; likewise, since A lies in that plane, A·L = 0.

An essential property of the LRL vector, which makes this conserved quantity unusual, is that for the three-dimensional Lagrangian of the system there does "not" exist a so-called cyclic coordinate corresponding to it, whereas for other conserved quantities it does. Thus the conservation of the LRL vector must be derived directly, e.g., by the method of Poisson brackets, as described below. Conserved quantities of this kind are called "dynamic", in contrast to the usual "geometric" conservation laws, e.g., that of the angular momentum.

History of rediscovery

The LRL vector A is a constant of motion of the important Kepler problem, and is useful in describing astronomical orbits, such as the motion of the planets. 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. Jakob Hermann was the first to show that A is conserved for a special case of the inverse-square central force, [cite journal| last = Hermann | first = J | authorlink = Jakob Hermann | year = 1710 | title = Unknown title | journal = Giornale de Letterati D'Italia | volume = 2 | pages = 447–467
cite journal| last = Hermann | first = J | authorlink = Jakob Hermann | year = 1710 | title = Extrait d'une lettre de M. Herman à M. Bernoulli datée de Padoüe le 12. Juillet 1710 | journal = Histoire de l'academie royale des sciences (Paris) | volume = 1732 | pages = 519–521
] 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. [cite journal| last = Bernoulli | first = J | authorlink = Johann Bernoulli | year = 1710 | title = Extrait de la Réponse de M. Bernoulli à M. Herman datée de Basle le 7. Octobre 1710 | journal = Histoire de l'academie royale des sciences (Paris) | volume = 1732 | pages = 521–544] At the end of the century, Pierre-Simon de Laplace rediscovered the conservation of A, deriving it analytically, rather than geometrically. [cite book | last = Laplace | first = PS | authorlink = Laplace | year = 1799 | title = Traité de mécanique celeste | pages = Tome I, Premiere Partie, Livre II, pp.165ff] In the middle of the nineteenth century, William Rowan Hamilton derived the equivalent eccentricity vector defined below, using it to show that the momentum vector p moves on a circle for motion under an inverse-square central force (Figure 3). At the beginning of the twentieth century, Josiah Willard Gibbs derived the same vector by vector analysis. [cite book | last = Gibbs | first = JW | authorlink = Josiah Willard Gibbs | coauthors = Wilson EB | year = 1901 | title = Vector Analysis | publisher = Scribners | location = New York | pages = p. 135] Gibbs' derivation was used as an example by Carle Runge in a popular German textbook on vectors, [cite book | last = Runge | first = C | authorlink = Carle David Tolme Runge | year = 1919 | title = Vektoranalysis | publisher = Hirzel | location = Leipzig | pages = Volume I] which was referenced by Wilhelm Lenz in his paper on the (old) quantum mechanical treatment of the hydrogen atom. [cite journal | last = Lenz | first = W | authorlink = Wilhelm Lenz | year = 1924 | title = Über den Bewegungsverlauf und Quantenzustände der gestörten Keplerbewegung | journal = Zeitschrift für Physik | volume = 24 | pages = 197–207 | doi = 10.1007/BF01327245] In 1926, the vector was used by Wolfgang Pauli to derive the spectrum of hydrogen using modern quantum mechanics, but not the Schrödinger equation; after Pauli's publication, it became known mainly as the "Runge–Lenz vector".

Mathematical definition

For a single particle acted on by an inverse-square central force described by the equation mathbf{F}(r)=frac{-k}{r^{2mathbf{hat{r, the LRL vector A is defined mathematically by the formula

: mathbf{A} = mathbf{p} imes mathbf{L} - m k mathbf{hat{r


* m!, is the mass of the point particle moving under the central force,
* mathbf{p}!, is its momentum vector,
* mathbf{L} = mathbf{r} imes mathbf{p}!, is its angular momentum vector,
* k!, is a parameter that describes strength of the central force,
* mathbf{r}!, is the position vector of the particle (Figure 1), and
* mathbf{hat{r!, is the corresponding unit vector, i.e., mathbf{hat{r = frac{mathbf{r{r} where "r" is the magnitude of r.

Since the assumed force is conservative, the total energy "E" is a constant of motion

:E = frac{p^{2{2m} - frac{k}{r} = frac{1}{2} mv^{2} - frac{k}{r}

Furthermore, the assumed force is a central force, and thus the angular momentum vector L is also conserved and defines the plane in which the particle travels. The LRL vector A is perpendicular to the angular momentum vector L because both p × L and r are perpendicular to L. It follows that A lies in the plane of the orbit.

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

A variety of alternative formulations for the same constant of motion may also be used. The most common is to scale by mk to define the eccentricity vector

:mathbf{e} = frac{mathbf{A{m k} = frac{1}{m k}(mathbf{p} imes mathbf{L}) - mathbf{hat{r

Derivation of the Kepler orbits

The "shape" and "orientation" of the Kepler problem orbits can be determined from the LRL vector as follows. Taking the dot product of A with the position vector r gives the equation

:mathbf{A} cdot mathbf{r} = Ar cos heta = mathbf{r} cdot left( mathbf{p} imes mathbf{L} ight) - mkr

where θ is the angle between r and A (Figure 2). Permuting the scalar triple product

:mathbf{r} cdotleft(mathbf{p} imes mathbf{L} ight) = left(mathbf{r} imes mathbf{p} ight)cdotmathbf{L} = mathbf{L}cdotmathbf{L}=L^2

and rearranging yields the defining formula for a conic section

:frac{1}{r} = frac{mk}{L^{2 left( 1 + frac{A}{mk} cos heta ight)

of eccentricity e!,

:e = frac{A}{mk} = frac{left|mathbf{A} ight{m k}

and latus rectum

:left| 4p ight| = frac{2L^{2{mk}

The major semiaxis "a" of the conic section may be defined using the latus rectum and the eccentricity

:a left( 1 pm e^{2} ight) = 2p = frac{L^{2{mk}

where the minus sign pertains to ellipses and the plus sign to hyperbolae.

Taking the dot product of A with itself yields an equation involving the energy "E"

:A^2= m^2 k^2 + 2 m E L^2 ,

which may be re-written in terms of the eccentricity

:e^{2} - 1= frac{2L^{2{mk^{2E

Thus, if the energy "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"), the eccentricity is greater than one and the orbit is a hyperbola. Finally, if the energy is exactly zero, the eccentricity is one and the orbit is a parabola. In all cases, the direction of 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.

Circular momentum hodographs

The conservation of the LRL vector A and angular momentum vector L is useful in showing that the momentum vector p moves on a circle under an inverse-square central force. Taking the cross product of A and L yields an equation for p

:L^{2} mathbf{p} = mathbf{L} imes mathbf{A} - mk hat{mathbf{r imes mathbf{L}

Taking L along the "z"-axis and the major semiaxis as the "x"-axis yields the equation

:p_{x}^{2} + left(p_{y} - A/L ight)^{2} = left( mk/L ight)^{2}

In other words, the momentum vector p is confined to a circle of radius "mk/L" centered on (0, "A/L"). The eccentricity "e" corresponds to the cosine of the angle η shown in Figure 3. For brevity, it is also useful to introduce the variable p_{0} = sqrt{2mleft| E ight. This circular hodograph is useful in illustrating the symmetry of the Kepler problem.

Constants of motion and superintegrability

The seven scalar quantities "E", A and L (being vectors, the latter two contribute three conserved quantities each) are related by two equations, A · L = 0 and "A2 = m2k2 + 2 m E L2", giving five independent constants of motion. 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. Since the magnitude of A (and the eccentricity "e" of the orbit) can be determined from the total angular momentum "L" and the energy "E", only the "direction" of A is conserved independently; moreover, since A must be perpendicular to L, it contributes only one additional conserved quantity.

A mechanical system with "d" degrees of freedom can have at most 2"d" − 1 constants of motion, since there are 2"d" initial conditions and the initial time cannot be determined by a constant of motion. A system with more than "d" constants of motion is called "superintegrable" and a system with 2"d" − 1 constants is called "maximally superintegrable". [cite journal | last = Evans | first = NW | year = 1990 | title = Superintegrability in classical mechanics | journal = Physical Review a | volume = 41 | pages = 5666–5676 | doi = 10.1103/PhysRevA.41.5666] Since the solution of the Hamilton–Jacobi equation in one coordinate system can yield only "d" constants of motion, superintegrable systems must be separable in more than one coordinate system. [cite book | last = Sommerfeld | first = A | authorlink = Arnold Sommerfeld | year = 1923 | title = Atomic Structure and Spectral Lines | publisher = Methuen | location = London | pages = 118] The Kepler problem is maximally superintegrable, since it has three degrees of freedom ("d=3") and five independent constant of motion; its Hamilton–Jacobi equation is separable in both spherical coordinates and parabolic coordinates,cite book | last=Landau |first=LD | authorlink=Lev Landau | coauthors=Lifshitz EM | year=1976 | title=Mechanics | edition=3rd edition | publisher=Pergamon Press | pages = p. 154 | id= ISBN 0-08-021022-8 (hardcover) and ISBN 0-08-029141-4 (softcover)] as described below. Maximally superintegrable systems follow closed, one-dimensional orbits in phase space, since the orbit is the intersection of the phase-space isosurfaces of their constants of motion. Maximally superintegrable systems can be quantized using only commutation relations, as illustrated below. [cite journal | last = Evans | first = NW | year = 1991 | title = Group theory of the Smorodinsky–Winternitz system | journal = Journal of Mathematical Physics | volume = 32 | pages = 3369–3375 | doi = 10.1063/1.529449]

Evolution under perturbed potentials

The Laplace–Runge–Lenz vector 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 "h"("r"). In such cases, the LRL vector rotates slowly in the plane of the orbit, corresponding to a slow precession of the orbit. By assumption, the perturbing potential "h"("r") is a conservative central force, which implies that the total energy "E" and angular momentum vector L are conserved. Thus, the motion still lies in a plane perpendicular to L and the magnitude "A" is conserved, from the equation "A"2 = "m"2"k"2+2"mEL"2. The perturbation potential "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 gives information about the perturbing potential "h(r)". Using canonical perturbation theory and action-angle coordinates, it is straightforward to show that A rotates at a rate of

:egin{array}{rcl}frac{partial}{partial L} langle h(r) angle & = &displaystyle frac{partial}{partial L} left{ frac{1}{T} int_{0}^{T} h(r) , dt ight} \ [1em] & = & displaystylefrac{partial}{partial L} left{ frac{m}{L^{2 int_{0}^{2pi} r^{2} h(r) , d heta ight}.end{array}

where "T" is the orbital period and the identity "L" "dt" = "m" "r"2 "d"θ was used to convert the time integral into an angular integral (Figure 5). The expression in angular brackets, 〈"h"("r")〉, represents the perturbing potential, but "averaged" over one full period; that is, averaged over one full passage of the body around its orbit. Mathematically, that 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 Einstein's theory of general relativity, which adds a small inverse-cubic perturbation to the normal Newtonian gravitational force.cite journal | last = Einstein | first = A | authorlink = Albert Einstein | year = 1915 | title = Erklärung der Perihelbewegung des Merkur aus der allgemeinen Relativitätstheorie | journal = Sitzungsberichte der Preussischen Akademie der Wissenschaften | volume = 1915 | pages = 831–839]

:h(r) = frac{kL^{2{m^{2}c^{2 left( frac{1}{r^{3 ight)

Inserting this function into the integral and using the equation

:frac{1}{r} = frac{mk}{L^{2 left( 1 + frac{A}{mk} cos heta ight)

to express "r" in terms of θ, the precession rate of the periapsis caused by this non-Newtonian perturbation is calculated to be

:frac{6pi k^{2{TL^{2}c^{2

which closely matches the observed anomalous precession of Mercury [cite journal | last = Le Verrier | first = UJJ | authorlink = Le Verrier | year = 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'Academie de Sciences (Paris) | volume = 49 | pages = 379–383] and binary pulsars. [cite book | last = Will | first = CM | year = 1979 | title = General Relativity, an Einstein Century Survey | edition = SW Hawking and W Israel, eds. | publisher = Cambridge University Press | location = Cambridge | pages = Chapter 2] This agreement with experiment is considered to be strong evidence for general relativity. [cite book | last = Pais | first = A. | authorlink = Abraham Pais | year = 1982 | title = Subtle is the Lord: The Science and the Life of Albert Einstein | publisher = Oxford University Press ] [cite book | last = Roseveare | first = NT | year = 1982 | title = Mercury's Perihelion from Le Verrier to Einstein | publisher = Oxford University Press]

Poisson brackets

The three components "Li" of the angular momentum vector L have the Poisson brackets

:left{ L_{i}, L_{j} ight} = sum_{s=1}^{3} epsilon_{ijs} L_{s}

where "i"=1,2,3 and εijs is the fully antisymmetric tensor, i.e., the Levi-Civita symbol; the summation index "s" is used here to avoid confusion with the force parameter "k" defined above. The Poisson brackets are represented here as "square" brackets (not curly braces), both for consistency with the references and because they will be interpreted as quantum mechanical commutation relations in the next section and as Lie brackets in a following section.

As noted above, a scaled Laplace–Runge–Lenz vector D may be defined with the same units as angular momentum by dividing A by "p0". The Poisson brackets of D with the angular momentum vector L can be written in a similar formcite book | last=Bohm | first=A. | year=1986 | title=Quantum Mechanics: Foundations and Applications | edition= 2nd edition | publisher=Springer Verlag | pages=208–222]

:left{ D_{i}, L_{j} ight} = sum_{s=1}^{3} epsilon_{ijs} D_{s}

The Poisson brackets of D with "itself" depend on the sign of "E", i.e., on whether the total energy "E" is negative (producing closed, elliptical orbits under an inverse-square central force) or positive (producing open, hyperbolic orbits under an inverse-square central force). For "negative" energies — i.e., for bound systems — the Poisson brackets are

:left{ D_{i}, D_{j} ight} = sum_{s=1}^{3} epsilon_{ijs} L_{s}

whereas, for "positive" energy, the Poisson brackets have the opposite sign

:left{ D_{i}, D_{j} ight} = -sum_{s=1}^{3} epsilon_{ijs} L_{s}

The Casimir invariants for negative energies are defined by

:C_{1} = mathbf{D} cdot mathbf{D} + mathbf{L} cdot mathbf{L} = frac{mk^{2{2left|E ight

:C_{2} = mathbf{D} cdot mathbf{L} = 0

and have zero Poisson brackets with all components of D and L

:left{ C_{1}, L_{i} ight} = left{ C_{1}, D_{i} ight} = left{ C_{2}, L_{i} ight} = left{ C_{2}, D_{i} ight} = 0

"C2" is trivially zero, since the two vectors are always perpendicular. However, the other invariant "C1" is non-trivial and depends only on "m", "k" and "E". This invariant allows the energy levels of hydrogen-like atoms to be derived using only quantum mechanical canonical commutation relations, instead of the more customary Schrödinger equation.

Quantum mechanics of the hydrogen atom

Poisson brackets provide a simple method for quantizing a classical system; the commutation relation of two quantum mechanical operators equals the Poisson bracket of the corresponding classical variables, multiplied by ihbar. [cite book | last = Dirac | first = PAM | authorlink = Paul Dirac | year = 1958 | title = Principles of Quantum Mechanics, 4th revised edition | publisher = Oxford University Press] By carrying out this quantization and calculating the eigenvalues of the C_{1} Casimir operator for the Kepler problem, Wolfgang Pauli was able to derive the energy levels of hydrogen-like atoms (Figure 6) and, thus, their atomic emission spectrum. This elegant derivation was obtained prior to the development of the Schrödinger equation. [cite journal | last = Schrödinger | first = E | authorlink = Erwin Schrödinger | year = 1926 | title = Quantisierung als Eigenwertproblem | journal = Annalen der Physik | volume = 384 | pages = 361–376 | doi = 10.1002/andp.19263840404]

A subtlety of the quantum mechanical operator for the LRL vector A is that the momentum and angular momentum operators do not commute; hence, the cross product of p and L must be defined carefully. Typically, the operators for the Cartesian components "As" are defined using a symmetric product

:A_{s} = - m k hat{r}_{s} + frac{1}{2} sum_{i=1}^{3} sum_{j=1}^{3} epsilon_{sij} left( p_{i} l_{j} + l_{j} p_{i} ight)

from which the corresponding ladder operators can be defined

:J_{0} = A_{3} ,

:J_{pm 1} = mp frac{1}{sqrt{2 left( A_{1} pm i A_{2} ight)

A normalized first Casimir invariant operator can likewise be defined

:C_{1} = - frac{m k^{2{2 hbar^{2 H^{-1} - I

where "H"−1 is the inverse of the Hamiltonian energy operator and "I" is the identity operator. Applying these ladder operators to the eigenstates left| l m n ight. angle of the total angular momentum, azimuthal angular momentum and energy operators, the eigenvalues of the first Casimir operator "C"1 are "n"2 − 1; importantly, they are independent of the "l" and "m" quantum numbers, making the energy levels degenerate. Hence, the energy levels are given by

:E_{n} = - frac{m k^{2{2hbar^{2} n^{2

which equals the Rydberg formula for hydrogen-like atoms (Figure 6).

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" electronic orbitals of the same energy, i.e., degenerate energy levels. A conserved quantity is usually associated with such symmetries. For example, every central force is symmetric under the rotation group SO(3), leading to the conservation of angular momentum L. Classically, an overall rotation of the system does not affect the energy of an orbit; quantum mechanically, rotations mix the spherical harmonics of the same quantum number "l" without changing the energy.

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 L and the LRL vector A (as defined above) and, quantum mechanically, ensures that the energy levels of hydrogen do not depend on the angular momentum quantum numbers "l" and "m". The symmetry is more subtle, however, because the symmetry operation must take place in a higher-dimensional space; such symmetries are often called "hidden symmetries". 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 "l" and "m" quantum numbers, such as the "s" ("l"=0) and "p" ("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 SO(4), which preserves the length of four-dimensional vectors

:left| mathbf{e} ight|^{2} = e_{1}^{2} + e_{2}^{2} + e_{3}^{2} + e_{4}^{2}

In 1935, Vladimir Fock showed that the quantum mechanical bound Kepler problem is equivalent to the problem of a free particle confined to a three-dimensional unit sphere in four-dimensional space. 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 reprojection results in a continuous mapping of the elliptical orbits without changing the energy; quantum mechanically, this corresponds to a mixing of all orbitals of the same energy quantum number "n". Valentine Bargmann noted subsequently that the Poisson brackets for the angular momentum vector L and the scaled LRL vector D formed the Lie algebra for SO(4). Simply put, the six quantities D and 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 — i.e., for unbound, "scattered" systems — the higher symmetry group is SO(3,1), which preserves the Minkowski length of 4-vectors

:ds^{2} = e_{1}^{2} + e_{2}^{2} + e_{3}^{2} - e_{4}^{2}

Both the negative- and positive-energy cases were considered by Fock and Bargmann and have been reviewed encyclopedically by Bander and Itzykson.cite journal | last = Bander | first = M | coauthors = Itzykson C | year = 1966 | title = Group Theory and the Hydrogen Atom (I) | journal = Reviews of Modern Physics | volume = 38 | pages = 330–345 | doi = 10.1103/RevModPhys.38.330] [cite journal | last = Bander | first = M | coauthors = Itzykson C | year = 1966 | title = Group Theory and the Hydrogen Atom (II) | journal = Reviews of Modern Physics | volume = 38 | pages = 346–358 | doi = 10.1103/RevModPhys.38.346]

The orbits of central-force systems — and those of the Kepler problem in particular — are also symmetric under reflection. Therefore, the SO(3), SO(4) and SO(3,1) groups cited above are not the full symmetry groups of their orbits; the full groups are O(3), O(4) and O(3,1), respectively. Nevertheless, only the connected subgroups, SO(3), SO(4) and SO(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

The connection between the Kepler problem and four-dimensional rotational symmetry SO(4) can be readily visualized.cite journal | last = Rogers | first = HH | year = 1973 | title = Symmetry transformations of the classical Kepler problem | journal = Journal of Mathematical Physics | volume = 14 | pages = 1125–1129 | doi = 10.1063/1.1666448] [cite book | last = Guillemin | first = V | coauthors = Sternberg S | year = 1990 | title = Variations on a Theme by Kepler | publisher = American Mathematical Society Colloquium Publications, volume 42 | id = ISBN 0-8218-1042-1] Let the four-dimensional Cartesian coordinates be denoted ("w", "x", "y", "z") where ("x", "y", "z") represent the Cartesian coordinates of the normal position vector r. The three-dimensional momentum vector p is associated with a four-dimensional vector oldsymboleta on a three-dimensional unit sphere

:egin{array}{rcl}oldsymboleta & = &displaystyle 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] & = & displaystyle frac{mk - r p_{0}^{2{mk} mathbf{hat{w + frac{rp_{0{mk} mathbf{p}end{array}

where mathbf{hat{w is the unit vector along the new "w"-axis. The transformation mapping p to η can be uniquely inverted; for example, the "x"-component of the momentum equals

:p_{x} = p_{0} frac{eta_{x{1 - eta_{w and similarly for "py" and "pz". In other words, the three-dimensional vector p is a stereographic projection of the four-dimensional oldsymboleta vector, scaled by "p"0 (Figure 8).

Without loss of generality, we may eliminate the normal rotational symmetry by choosing the Cartesian coordinates such that the "z"-axis is aligned with the angular momentum vector L and the momentum hodographs are aligned as they are in Figure 7, with the centers of the circles on the "y"-axis. Since the motion is planar, and p and L are perpendicular, "pz" = η"z" = 0 and attention may be restricted to the three-dimensional vector oldsymboleta = (η"w", η"x", η"y"). The family of Apollonian circles of momentum hodographs (Figure 7) correspond to a family of great circles on the three-dimensional oldsymboleta sphere, all of which intersect the η"x"-axis at the two foci "ηx" = ±1, corresponding to the momentum hodograph foci at "px" = ±"p"0. These great circles are related by a simple rotation about the η"x"-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 η"w". 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 oldsymboletain favor of elliptic cylindrical coordinates (χ, ψ, φ) [cite journal | last = Lakshmanan | first = M | coauthors = Hasegawa H | title = On the canonical equivalence of the Kepler problem in coordinate and momentum spaces | journal = Journal of Physics a | volume = 17 | pages = L889–L893 | doi = 10.1088/0305-4470/17/16/006 | year = 1984]

:eta_{w} = mathrm{cn}, chi mathrm{cn}, psi:eta_{x} = mathrm{sn}, chi mathrm{dn}, psi cos phi:eta_{y} = mathrm{sn}, chi mathrm{dn}, psi sin phi:eta_{z} = mathrm{dn}, chi mathrm{sn}, psi

where sn, cn and 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 an electric field E, the conserved generalized Laplace–Runge–Lenz vector mathcal{A} is [cite journal | last = Redmond | first = PJ | year = 1964 | title = Generalization of the Runge–Lenz Vector in the Presence of an Electric Field | journal = Physical Review | volume = 133 | pages = B1352–B1353 | doi = 10.1103/PhysRev.133.B1352]

:mathcal{A} = mathbf{A} + frac{mq}{2} left [ left( mathbf{r} imes mathbf{E} ight) imes mathbf{r} ight]

where "q" is the charge on the orbiting particle.

Further generalizing the Laplace–Runge–Lenz vector to other potentials and special relativity, the most general form can be written as

:mathcal{A} = left( frac{partial xi}{partial u} ight) left(mathbf{p} imes mathbf{L} ight) + left [ xi - u left( frac{partial xi}{partial u} ight) ight] L^{2} mathbf{hat{r

where "u" = "1/r" (cf. Bertrand's theorem) and ξ = cos θ, with the angle θ defined by

: heta = L int^{u} frac{du}{sqrt{m^{2} c^{2} left(gamma^{2} - 1 ight) - L^{2} u^{2}

and γ is the Lorentz factor. As before, we may obtain a conserved binormal vector B by taking the cross product with the conserved angular momentum vector

:mathcal{B} = mathbf{L} imes mathcal{A}

These two vectors may likewise be combined into a conserved dyadic tensor W

:mathcal{W} = alpha mathcal{A} otimes mathcal{A} + eta , mathcal{B} otimes mathcal{B}

For illustration, the LRL vector for a non-relativistic, isotopic harmonic oscillator can be calculated. Since the force is central

:mathbf{F}(r)= -k mathbf{r}

the angular momentum vector is conserved and the motion lies in a plane. The conserved dyadic tensor can be written in a simple form

:mathcal{W} = frac{1}{2m} mathbf{p} otimes mathbf{p} + frac{k}{2} , mathbf{r} otimes mathbf{r}

although it should be noted that p and r are not necessarily perpendicular. The corresponding Runge–Lenz vector is more complicated

:mathcal{A} = frac{1}{sqrt{mr^{2}omega_{0} A - mr^{2}E + L^{2} left{ left( mathbf{p} imes mathbf{L} ight) + left(mromega_{0} A - mrE ight) mathbf{hat{r ight}

where omega_{0} = sqrt{frac{k}{m is the natural oscillation frequency.

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 mathbf{F} acting on the particle is

:mathbf{F} = frac{dmathbf{p{dt} = f(r) frac{mathbf{r{r} = f(r) mathbf{hat{r

for some function f(r) of the radius r. Since the angular momentum mathbf{L} = mathbf{r} imes mathbf{p} is conserved under central forces, frac{d}{dt}mathbf{L} = 0 and

:frac{d}{dt} left( mathbf{p} imes mathbf{L} ight) = frac{dmathbf{p{dt} imes mathbf{L} = f(r) mathbf{hat{r imes left( mathbf{r} imes m frac{dmathbf{r{dt} ight) = f(r) frac{m}{r} left [ mathbf{r} left(mathbf{r} cdot frac{dmathbf{r{dt} ight) - r^{2} frac{dmathbf{r{dt} ight]

where the momentum mathbf{p} = m frac{dmathbf{r{dt} and where the triple cross product has been simplified using Lagrange's formula

:mathbf{r} imes left( mathbf{r} imes frac{dmathbf{r{dt} ight) = mathbf{r} left(mathbf{r} cdot frac{dmathbf{r{dt} ight) - r^{2} frac{dmathbf{r{dt}

The identity

:frac{d}{dt} left( mathbf{r} cdot mathbf{r} ight) = 2 mathbf{r} cdot frac{dmathbf{r{dt} = frac{d}{dt} left( r^{2} ight) = 2rfrac{dr}{dt}

yields the equation

:frac{d}{dt} left( mathbf{p} imes mathbf{L} ight) = -m f(r) r^{2} left [ frac{1}{r} frac{dmathbf{r{dt} - frac{mathbf{r{r^{2 frac{dr}{dt} ight] = -m f(r) r^{2} frac{d}{dt} left( frac{mathbf{r{r} ight)

For the special case of an inverse-square central force f(r)=frac{-k}{r^{2, this equals

:frac{d}{dt} left( mathbf{p} imes mathbf{L} ight) = m k frac{d}{dt} left( frac{mathbf{r{r} ight) = frac{d}{dt} left( mkmathbf{hat{r ight)

Therefore, A is conserved for inverse-square central forces

:frac{d}{dt} mathbf{A} = frac{d}{dt} left( mathbf{p} imes mathbf{L} ight) - frac{d}{dt} left( mkmathbf{hat{r ight) = 0

As described below, this LRL vector A is a special case of a general conserved vector mathcal{A} that can be defined for all central forces. However, since most central forces do not produce closed orbits (see Bertrand's theorem), the analogous vector mathcal{A} rarely has a simple definition and is generally a multivalued function of the angle θ between r and mathcal{A}.

Hamilton–Jacobi equation in parabolic coordinates

The constancy of the LRL vector can also be derived from the Hamilton–Jacobi equation in parabolic coordinates (ξ, η), which are defined by the equations

:xi = r + x ,

:eta = r - x ,

where "r" represents the radius in the plane of the orbit

:r = sqrt{x^{2} + y^{2

The inversion of these coordinates is

:x = frac{1}{2} left( xi - eta ight)

:y = sqrt{xieta}

Separation of the Hamilton–Jacobi equation in these coordinates yields the two equivalent equations [cite journal | last = Dulock | first = VA | coauthors = McIntosh HV | year = 1966 | title = On the Degeneracy of the Kepler Problem | journal = Pacific Journal of Mathematics | volume = 19 | pages = 39–55]

:2xi p_{xi}^{2} - mk - mExi = -Gamma

:2eta p_{eta}^{2} - mk - mEeta = Gamma

where Γ is a constant of motion. Subtraction and re-expression in terms of the Cartesian momenta "px" and "py" shows that Γ is equivalent to the LRL vector

:Gamma = p_{y} left( x p_{y} - y p_{x} ight) - mkfrac{x}{r} = A_{x}

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 coordinates of a physical system

:delta q_{i} = epsilon g_{i}(mathbf{q}, mathbf{dot{q, t)

that causes the Lagrangian to vary to first order by a total time derivative

:delta L = epsilon frac{d}{dt} G(mathbf{q}, t)

corresponds to a conserved quantity Γ

:Gamma = -G + sum_{i} g_{i} left( frac{partial L}{partial dot{q}_{i ight)

In particular, the conserved LRL vector component "As" corresponds to the variation in the coordinates [cite journal | last = Lévy-Leblond | first = JM | year = 1971 | title = Conservation Laws for Gauge-Invariant Lagrangians in Classical Mechanics | journal = American Journal of Physics | volume = 39 | pages = 502–506 | doi = 10.1119/1.1986202]

:delta x_{i} = frac{epsilon}{2} left [ 2 p_{i} x_{s} - x_{i} p_{s} - delta_{is} left( mathbf{r} cdot mathbf{p} ight) ight]

where "i" equals 1, 2 and 3, with "xi" and "pi" being the "i"th components of the position and momentum vectors r and p, respectively; as usual, "δis" represents the Kronecker delta. The resulting first-order change in the Lagrangian is

:delta L = epsilon mkfrac{d}{dt} left( frac{x_{s{r} ight)

Substitution into the general formula for the conserved quantity Γ yields the conserved component "As" of the LRL vector

:A_{s} = left [ p^{2} x_{s} - p_{s} left(mathbf{r} cdot mathbf{p} ight) ight] - mk left( frac{x_{s{r} ight) = left [ mathbf{p} imes mathbf{r} imes mathbf{p} ight] _{s} - mk left( frac{x_{s{r} ight)

Lie transformation

The Noether theorem derivation of the conservation of the LRL vector A is elegant, but has one drawback: the coordinate variation δ"x"i involves not only the "position" r, but also the "momentum" p or, equivalently, the "velocity" v. [cite journal | last = Gonzalez-Gascon | first = F | year = 1977 | title = Notes on the symmetries of systems of differential equations | journal = Journal of Mathematical Physics | volume = 18 | pages = 1763–1767 | doi = 10.1063/1.523486] This drawback may be eliminated by instead deriving the conservation of A using an approach pioneered by Sophus Lie. [cite book | last = Lie | first = S | authorlink = Sophus Lie | year = 1891 | title = Vorlesungen über Differentialgleichungen | publisher = Teubner | location = Leipzig] [cite book | last = Ince | first = EL | year = 1926 | title = Ordinary Differential Equations | publisher = Dover (1956 reprint) | location = New York | pages = 93–113] Specifically, one may define a Lie transformationcite journal | last = Prince | first = GE | coauthors = Eliezer CJ | year = 1981 | title = On the Lie symmetries of the classical Kepler problem | journal = Journal of Physics A: Mathematical and General | volume = 14 | pages = 587–596 | doi = 10.1088/0305-4470/14/3/009] in which the coordinates r and the time "t" are scaled by different powers of a parameter λ (Figure 9)

:t ightarrow lambda^{3}t, mathbf{r} ightarrow lambda^{2}mathbf{r}, mathbf{p} ightarrow frac{1}{lambda}mathbf{p}

This transformation changes the total angular momentum "L" and energy "E"

:L ightarrow lambda L, E ightarrow frac{1}{lambda^{2 E

but preserves their product "EL2". Therefore, the eccentricity "e" and the magnitude "A" are preserved, as may be seen from the equation for "A"2

:A^2 = m^2 k^2 e^{2} = m^2 k^2 + 2 m E L^2

The direction of 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 "a" and the period "T" form a constant "T2/a3".

Alternative scalings, symbols and formulations

Unlike the momentum and angular momentum vectors p and 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 above, but another common alternative is to divide by the constant "mk" to obtain a dimensionless conserved eccentricity vector

: mathbf{e} = frac{1}{mk} left(mathbf{p} imes mathbf{L} ight) - mathbf{hat{r = frac{m}{k} left(mathbf{v} imes mathbf{r} imes mathbf{v} ight) - mathbf{hat{r

where v is the velocity vector. This scaled vector e has the same direction as A and its magnitude equals the eccentricity of the orbit. Other scaled versions are also possible, e.g., by dividing A by "m" alone

: mathbf{M} = mathbf{v} imes mathbf{L} - kmathbf{hat{r

or by "p"0

:mathbf{D} = frac{mathbf{A{p_{0 = frac{1}{sqrt{2mleft| E ight| left{ mathbf{p} imes mathbf{L} - m k mathbf{hat{r ight}

which has the same units as the angular momentum vector L. In rare cases, the sign of the LRL vector may be reversed, i.e., scaled by −1. Other common symbols for the LRL vector include a, R, F, J and V. However, the choice of scaling and symbol for the LRL vector do not affect its conservation.

An alternative conserved vector is the binormal vector B studied by William Rowan Hamilton

:mathbf{B} = mathbf{p} - left(frac{mk}{L^{2}r} ight) left( mathbf{L} imes mathbf{r} ight)

which is conserved and points along the "minor" semiaxis of the ellipse; the LRL vector A = B × L is the cross product of B and L (Figure 4). The vector B is denoted as "binormal" since it is perpendicular to both A and L. Similar to the LRL vector itself, the binormal vector can be defined with different scalings and symbols.

The two conserved vectors, A and B can be combined to form a conserved dyadic tensor W

:mathbf{W} = alpha mathbf{A} otimes mathbf{A} + eta , mathbf{B} otimes mathbf{B}

where α and β are arbitrary scaling constants and otimes represents the tensor product (which is not related to the vector cross product, despite their similar symbol). Written in explicit components, this equation reads

:W_{ij} = alpha A_{i} A_{j} + eta B_{i} B_{j} ,

Being perpendicular to each another, the vectors A and B can be viewed as the principal axes of the conserved tensor W, i.e., its scaled eigenvectors. W is perpendicular to L

:mathbf{L} cdot mathbf{W} = alpha left( mathbf{L} cdot mathbf{A} ight) mathbf{A} + eta left( mathbf{L} cdot mathbf{B} ight) mathbf{B} = 0

since A and B are both perpendicular to L as well, LA = LB = 0. For clarification, this equation reads in explicit components

:left( mathbf{L} cdot mathbf{W} ight)_{j} = alpha left( sum_{i=1}^{3} L_{i} A_{i} ight) A_{j} + eta left( sum_{i=1}^{3} L_{i} B_{i} ight) B_{j} = 0

ee also

*Two-body problem
*Bertrand's theorem
*Quantum mechanics
*Astrodynamics: Orbit, Eccentricity vector, Orbital elements


Further reading

* | doi=10.2991/jnmp.2003.10.3.6

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