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Throughout this article, vectors and their magnitudes are indicated by boldface and italic type, respectively, e.g., <math>\left| \mathbf{A} \right| = A</math>.

The Laplace-Runge-Lenz vector is a constant of motion in key problems of classical mechanics, most notably Kepler's problem:<ref>Arnold, VI (1989). Mathematical Methods of Classical Mechanics, 2nd ed.. New York: Springer-Verlag, 38. ISBN 0-387-96890-3.</ref> determining the motion of two bodies interacting by a central force that varies as the inverse square of the distance between them (such as Newtonian gravity or Coulomb's law of electrostatics). If this vector is given, the shape of their relative motion can be deduced by simple geometry.<ref name="goldstein_1980">Goldstein, H. (1980). Classical Mechanics, 2^nd edition, Addison Wesley, 102–105,421–422.</ref>

By the correspondence principle, the Laplace-Runge-Lenz vector has a quantum mechanical analogue, which was critical in the first derivation of the spectrum of the hydrogen atom,<ref name="pauli_1926">Pauli, W (1926). "Über das Wasserstoffspektrum vom Standpunkt der neuen Quantenmechanik". Zeitschrift für Physik 36: 336–363.</ref> before the invention of the Schrödinger equation.

The Kepler problem has the unusual property that the momentum vector p always traces out a circle.<ref name="hamilton_1847_hodograph">Hamilton, WR (1847). "Unknown title". Proceedings of the Royal Irish Academy 3: 344ff.</ref> Due to the arrangement of these circles for a given total energy E, the Kepler problem is mathematically equivalent to a particle moving freely on a four-dimensional sphere.<ref name="fock_1935" >Fock, V (1935). "Zur Theorie des Wasserstoffatoms". Zeitschrift für Physik 98: 145–154.</ref> By this mathematical analogy, the conserved Laplace-Runge-Lenz vector is equivalent to extra components of the angular momentum in the four-dimensional space.<ref name="bargmann_1936" >Bargmann, V (1936). "Zur Theorie des Wasserstoffatoms: Bemerkungen zur gleichnamigen Arbeit von V. Fock". Zeitschrift für Physik 99: 576–582.</ref>

The Laplace-Runge-Lenz vector is also known as the Laplace vector, the Runge-Lenz vector and the Lenz vector although, ironically, none of those scientists invented it. The Laplace-Runge-Lenz vector has been re-discovered several times<ref name="goldstein_1975_1976">Goldstein, H. (1975). "Prehistory of the Runge-Lenz vector". American Journal of Physics 43: 735–738.
Goldstein, H. (1976). "More on the prehistory of the Runge-Lenz vector". American Journal of Physics 44: 1123–1124.</ref> and is also equivalent to the dimensionless eccentricity vector of celestial mechanics.<ref name="hamilton_1847_quaternions">Hamilton, WR (1847). "Applications of Quaternions to Some Dynamical Questions". Proceedings of the Royal Irish Academy 3: Appendix III.</ref> Similarly, there is no consensus for its symbol, although A is used most commonly.

Contents 1 Context and overview 2 History of rediscovery 3 Basic properties and applications 4 Constants of motion and superintegrability 5 Alternative scalings, symbols and formulations 6 Conservation under inverse-square forces 7 Evolution under perturbing potentials 8 Poisson brackets 9 Quantum mechanics of the hydrogen atom 10 Hamilton-Jacobi equation in parabolic coordinates 11 Noether's theorem 12 Lie transformation 13 Conservation and symmetry 14 Rotational symmetry in four dimensions 15 Generalizations to other potentials and relativity 16 See also 17 References 18 Further reading

[edit] Context and overview

A single particle moving under any conservative central force has at least four constants of motion, the total energy E and the angular momentum vector L. Qualitatively, 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 (Figure 1). This plane is perpendicular to the constant vector L, which may be expressed mathematically by the vector dot product equation r•L = 0.

As defined below, the Laplace-Runge-Lenz vector A always lies in the plane of motion — i.e., A•L = 0 — for any central force. However, A is constant only for an inverse-square central force.<ref name="goldstein_1980" /> If the central force is approximately an inverse-square law, the vector A is approximately constant in length, but slowly rotates its direction. For most central forces, however, the vector A — as defined below — is not constant, but changes in both length and direction. A generalized Laplace-Runge-Lenz vector <math>\mathcal{A}</math>, which is conserved, is defined below 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">Fradkin, DM (1967). "Existence of the Dynamic Symmetries O₄ and SU₃ for All Classical Central Potential Problems". Progress of Theoretical Physics 37: 798–812.</ref><ref name="yoshida_1987">Yoshida, T (1987). "Two methods of generalisation of the Laplace-Runge-Lenz vector". European Journal of Physics 8: 258–259.</ref>

The Laplace-Runge-Lenz vector A is defined below for a single point particle of mass m moving in a fixed potential. However, it may also 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.

[edit] History of rediscovery

The Laplace-Runge-Lenz 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 (anschaulich) 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 A is conserved for a special case of the inverse-square central force,<ref>Hermann, J (1710). "Unknown title". Giornale de Letterati D'Italia 2: 447–467.
Hermann, J (1710). "Extrait d'une lettre de M. Herman à M. Bernoulli datée de Padoüe le 12. Juillet 1710". Histoire de l'academie royale des sciences (Paris) 1732: 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>Bernoulli, J (1710). "Extrait de la Réponse de M. Bernoulli à M. Herman datée de Basle le 7. Octobre 1710". Histoire de l'academie royale des sciences (Paris) 1732: 521–544.</ref> At the end of the century, Pierre-Simon de Laplace rediscovered the conservation of A, deriving it analytically, rather than geometrically.<ref>Laplace, PS (1799). Traité de mécanique celeste, Tome I, Premiere Partie, Livre II, pp.165ff.</ref> In the middle of the nineteenth century, William Rowan Hamilton derived the equivalent eccentricity vector defined below,<ref name="hamilton_1847_quaternions" /> using it to show that the momentum vector p moves on a circle for motion under an inverse-square central force (Figure 2).<ref name="hamilton_1847_hodograph" /> At the beginning of the twentieth century, Josiah Willard Gibbs derived the same vector by vector analysis.<ref>Gibbs, JW, Wilson EB (1901). Vector Analysis. New York: Scribners, p. 135.</ref> Gibbs' derivation was used as an example by Carle Runge in a popular German textbook on vectors,<ref>Runge, C (1919). Vektoranalysis. Leipzig: Hirzel, Volume I.</ref> which was referenced by Wilhelm Lenz in his paper on the (old) quantum mechanical treatment of the hydrogen atom.<ref>Lenz, W (1924). "Über den Bewegungsverlauf und Quantenzustände der gestörten Keplerbewegung". Zeitschrift für Physik 24: 197–207.</ref> In 1926, the vector was used by Wolfgang Pauli to derive the spectrum of hydrogen using modern quantum mechanics, but not the Schrödinger equation;<ref name="pauli_1926" /> after Pauli's publication, it became known mainly as the Runge-Lenz vector.

[edit] Basic properties and applications

For a single particle acted on by an inverse-square central force <math>\mathbf{F}(r)=\frac{-k}{r^{2}}\mathbf{\hat{r}}</math>, the Laplace-Runge-Lenz vector A is defined mathematically by the equation<ref name="goldstein_1980" />

<math>

\mathbf{A} = \mathbf{p} \times \mathbf{L} - m k \frac{\mathbf{r}}{r} = \mathbf{p} \times \mathbf{L} - m k \mathbf{\hat{r}} </math>

where

• <math>m\!\,</math> is the mass of the point particle moving under the central force,
• <math>\mathbf{p}\!\,</math> is its momentum vector,
• <math>\mathbf{L} = \mathbf{r} \times \mathbf{p}\!\,</math> is its angular momentum vector,
• <math>k\!\,</math> is a parameter that describes strength of the central force,
• <math>\mathbf{r}\!\,</math> is the position vector of the particle, and
• <math>\mathbf{\hat{r}}\!\,</math> is the corresponding unit vector, i.e., <math>\mathbf{\hat{r}} = \frac{\mathbf{r}}{r}</math> where r is the magnitude of r.

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

<math>

E = \frac{p^{2}}{2m} - \frac{k}{r} = \frac{1}{2} mv^{2} - \frac{k}{r} </math>

Moreover, since the assumed force is a central force, the angular momentum vector L is conserved and defines the plane in which the particle travels. The Laplace-Runge-Lenz vector A is perpendicular to the angular momentum vector L and, thus, lies in the plane of the orbit. The equation A•L = 0 holds because p×L and r are both perpendicular to L; the definition L=r×p implies that r•L = 0.

The shape and orientation of the Kepler problem orbits can likewise be determined from the Laplace-Runge-Lenz vector. Taking the dot product of A with the position vector r gives the equation

<math>

\mathbf{A} \cdot \mathbf{r} = Ar \cos\theta = \mathbf{r} \cdot \left( \mathbf{p} \times \mathbf{L} \right) - mkr </math>

where θ is the angle between r and A. Permuting the scalar triple product r•(p×L)=L•(r×p)=L•L=L², and rearranging yields the defining formula for a conic section

<math>

\frac{1}{r} = \frac{mk}{L^{2}} \left( 1 + \frac{A}{mk} \cos\theta \right) </math>

of eccentricity <math>e\!\,</math> given by

<math>

e = \frac{A}{mk} = \frac{\left|\mathbf{A}\right|}{m k} </math>

Taking the dot product of A with itself yields the equation

<math>

A^2= m^2 k^2 + 2 m E L^2 </math>

which may be re-written in terms of the eccentricity

<math>

e^{2} - 1= \frac{2L^{2}}{mk^{2}}E </math>

Thus, if the energy 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.

The conservation of the Laplace-Runge-Lenz 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

<math>

L^{2} \mathbf{p} = \mathbf{L} \times \mathbf{A} - mk \hat{\mathbf{r}} \times \mathbf{L} </math>

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

<math>

p_{x}^{2} + \left(p_{y} - A/L \right)^{2} = \left( mk/L \right)^{2} </math>

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 2. For brevity, it is also useful to introduce the variable <math>p_{0} = \sqrt{2m\left| E \right|}</math>. This circular hodograph is useful in illustrating the symmetry of the Kepler problem.

[edit] Constants of motion and superintegrability

The seven scalar quantities E, A and L (being vectors, the latter two count as three conserved quantities) 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 of the orbit) can be determined from the total angular momentum L and the energy E, it could be argued that 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 2d-1 constants of motion, since there are 2d 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 2d-1 constants is called maximally superintegrable.<ref>Evans, NW (1990). "Superintegrability in classical mechanics". Physical Review A 41: 5666–5676.</ref> 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.<ref>Sommerfeld, A (1923). Atomic Structure and Spectral Lines. London: Methuen, 118.</ref> 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,<ref name="landau_lifshitz_1976">Landau, LD, Lifshitz EM (1976). Mechanics, 3^rd edition, Pergamon Press, p. 154. ISBN 0-08-021022-8 (hardcover) and ISBN 0-08-029141-4 (softcover).</ref> 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.<ref>Evans, NW (1991). "Group theory of the Smorodinsky-Winternitz system". Journal of Mathematical Physics 32: 3369–3375.</ref>

[edit] Alternative scalings, symbols and formulations

Contrary to 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

<math>

\mathbf{e} = \frac{1}{mk} \left(\mathbf{p} \times \mathbf{L} \right) - \mathbf{\hat{r}} = \frac{m}{k} \left(\mathbf{v} \times \mathbf{r} \times \mathbf{v}\right) - \mathbf{\hat{r}} </math>

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

<math>

\mathbf{M} = \mathbf{v} \times \mathbf{L} - k\mathbf{\hat{r}} </math>

or by p₀

<math>

\mathbf{D} = \frac{\mathbf{A}}{p_{0}} = \frac{1}{\sqrt{2m\left| E \right|}} \left\{ \mathbf{p} \times \mathbf{L} - m k \mathbf{\hat{r}} \right\} </math>

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

An alternative conserved vector is the binormal vector B studied by William Rowan Hamilton<ref name="hamilton_1847_quaternions" />

<math>

\mathbf{B} = \mathbf{p} - \left(\frac{mk}{L^{2}r} \right) \ \left( \mathbf{L} \times \mathbf{r} \right) </math>

which is conserved and points along the minor semiaxis of the ellipse; the Laplace-Runge-Lenz vector A = B×L is the cross product of B and L (Figure 3). The vector B is denoted as "binormal" since it is perpendicular to both A and L. Similar to the Laplace-Runge-Lenz 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

<math>

\mathbf{W} = \alpha \mathbf{A} \otimes \mathbf{A} + \beta \, \mathbf{B} \otimes \mathbf{B} </math>

where <math>\otimes</math> is the tensor product and α and β are arbitrary scaling constants.<ref name="fradkin_1967" /> Written in explicit components, this equation reads

<math>

W_{ij} = \alpha A_{i} A_{j} + \beta B_{i} B_{j} </math>

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

<math>

\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 A and B are similarly perpendicular, <math>\mathbf{L} \cdot \mathbf{A} = \mathbf{L} \cdot \mathbf{B} = 0</math>.

[edit] Conservation under inverse-square forces

The force <math>\mathbf{F}</math> acting on the particle is assumed to be a central force

<math>

\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>\frac{d}{dt}\mathbf{L} = 0</math> and

<math>

\frac{d}{dt} \left( \mathbf{p} \times \mathbf{L} \right) = \frac{d\mathbf{p}}{dt} \times \mathbf{L} = f(r) \mathbf{\hat{r}} \times \mathbf{r} \times m \frac{d\mathbf{r}}{dt} = 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>\mathbf{p} = m \frac{d\mathbf{r}}{dt}</math> and where the triple cross product has been simplified using Lagrange's formula

<math>

\mathbf{r} \times \mathbf{r} \times \frac{d\mathbf{r}}{dt} = \mathbf{r} \left(\mathbf{r} \cdot \frac{d\mathbf{r}}{dt} \right) - r^{2} \frac{d\mathbf{r}}{dt} </math>

The identity

<math>

\frac{d}{dt} \left( \mathbf{r} \cdot \mathbf{r} \right) = 2 \mathbf{r} \cdot \frac{d\mathbf{r}}{dt} = \frac{d}{dt} \left( r^{2} \right) = 2r\frac{dr}{dt} </math>

yields the equation

<math>

\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>f(r)=\frac{-k}{r^{2}}</math>, this equals

<math>

\frac{d}{dt} \left( \mathbf{p} \times \mathbf{L} \right) = m k \frac{d}{dt} \left( \frac{\mathbf{r}}{r}\right) = \frac{d}{dt} \left( mk\mathbf{\hat{r}} \right) </math>

Therefore, A is conserved for inverse-square central forces

<math>

\frac{d}{dt} \mathbf{A} = \frac{d}{dt} \left( \mathbf{p} \times \mathbf{L} \right) - \frac{d}{dt} \left( mk\mathbf{\hat{r}} \right) = 0 </math>

Analogous conserved vectors can be defined for all other central forces, but none are as simple as A.<ref name="fradkin_1967" /><ref name="yoshida_1987" /> Since most central forces do not produce closed orbits (see Bertrand's theorem), these constant vectors are multivalued functions of the angle θ.

[edit] Evolution under perturbing potentials

In many practical problems such as planetary motion, the interaction between two bodies is approximately an inverse-square central force, but not exactly. In such cases, the Laplace-Runge-Lenz vector A is not perfectly constant. However, if the perturbing potential h(r) is a conservative central force, the total energy E and angular momentum vector L are conserved. Therefore, the motion still lies in a plane perpendicular to L and the magnitude A is conserved, by the equation <math>A^2= m^2 k^2 +2 m E L^2 \!\,</math>. Consequently, the direction of A slowly rotates in the orbit of the plane; using canonical perturbation theory and action-angle coordinates, it is straightforward to show that A rotates at a rate of

<math>

\frac{\partial}{\partial L} \langle h(r) \rangle = \frac{\partial}{\partial L} \left\{ \frac{1}{T} \int_{0}^{T} h(r) \ dt \right\} = \frac{\partial}{\partial L} \left\{ \frac{m}{L^{2}} \int_{0}^{2\pi} r^{2} h(r) d\theta \right\} </math>

where T is the orbital period and the identity L dt = m r² dθ was used to convert the time integral into an angular integral (Figure 4). As an example, the theory of general relativity adds a small inverse-cubic perturbation to the normal Newtonian inverse-square gravitational force; specifically,

<math>

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>

\frac{1}{r} = \frac{mk}{L^{2}} \left( 1 + \frac{A}{mk} \cos\theta \right) </math>

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

<math>

\frac{6\pi k^{2}}{TL^{2}c^{2}} </math>

which closely matches the observed orbital precession of Mercury and binary pulsars. This agreement with experiment is considered to be strong evidence for general relativity.

[edit] Poisson brackets

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

<math>

\left[ L_{i}, L_{j}\right] = \sum_{s=1}^{3} \epsilon_{ijs} L_{s} </math>

where i=1,2,3 and ε_ijs 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 form

<math>

\left[ D_{i}, L_{j}\right] = \sum_{s=1}^{3} \epsilon_{ijs} D_{s} </math>

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

<math>

\left[ D_{i}, D_{j}\right] = \sum_{s=1}^{3} \epsilon_{ijs} L_{s} </math>

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

<math>

\left[ D_{i}, D_{j}\right] = -\sum_{s=1}^{3} \epsilon_{ijs} L_{s} </math>

The Casimir invariants for negative energies are defined by

<math>

C_{1} = \mathbf{D} \cdot \mathbf{D} + \mathbf{L} \cdot \mathbf{L} = \frac{mk^{2}}{2\left|E\right|} </math>

<math>

C_{2} = \mathbf{D} \cdot \mathbf{L} = 0 </math>

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

<math>

\left[ C_{1}, L_{i} \right] = \left[ C_{1}, D_{i} \right] = \left[ C_{2}, L_{i} \right] = \left[ C_{2}, D_{i} \right] = 0 </math>

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.

[edit] 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 <math>i\hbar</math>.<ref>Dirac, PAM (1958). Principles of Quantum Mechanics, 4th revised edition. Oxford University Press.</ref> By carrying out this quantization and calculating the eigenvalues of the <math>C_{1}</math> Casimir operator for the Kepler problem, Wolfgang Pauli was able to derive the energy levels of hydrogen-like atoms and, thus, their atomic emission spectrum.<ref name="pauli_1926" /> This elegant derivation was obtained prior to the development of the Schrödinger equation.<ref>Schrödinger, E (1926). "Quantisierung als Eigenwertproblem". Annalen der Physik 384: 361–376.</ref>

A subtlety of the quantum mechanical operator for the Laplace-Runge-Lenz 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.<ref>Bohm, A. (1986). Quantum Mechanics: Foundations and Applications, 2^nd edition, Springer Verlag, 208–222.</ref> Typically, the operators for the Cartesian components A_s are defined using a symmetric product

<math>

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} \right) </math>

from which the corresponding ladder operators can be defined

<math>

A_{0} = A_{3} </math>

<math>

A_{\pm 1} = \mp \frac{1}{\sqrt{2}} \left( A_{1} \pm i A_{2} \right) </math>

A normalized first Casimir invariant operator can likewise be defined

<math>

C_{1} = - \frac{m k^{2}}{2 \hbar^{2}} H^{-1} - I </math>

where H^-1 is the inverse of the Hamiltonian energy operator and I is the identity operator. Applying these ladder operators to the eigenstates <math>\left| l m n \right.\rangle</math> of the total angular momentum, azimuthal angular momentum and energy operators, it can be shown that the eigenstates of the first Casimir operator are n² - 1. Hence, the energy levels are given by

<math>

E_{n} = - \frac{m k^{2}}{2\hbar^{2} n^{2}} </math>

which equals the Rydberg formula for hydrogen-like atoms (Figure 5).

[edit] Hamilton-Jacobi equation in parabolic coordinates

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

<math>

\xi = r + x </math>

<math>

\eta = r - x </math>

where r represents the radius in the plane of the orbit

<math>

r = \sqrt{x^{2} + y^{2}} </math>

The inversion of these coordinates is

<math>

x = \frac{1}{2} \left( \xi - \eta \right) </math>

<math>

y = \sqrt{\xi\eta} </math>

Separation of the Hamilton-Jacobi equation in these coordinates yields the two equivalent equations<ref name="landau_lifshitz_1976" /><ref>Dulock, VA, McIntosh HV (1966). "On the Degeneracy of the Kepler Problem". Pacific Journal of Mathematics 19: 39–55.</ref>

<math>

2\xi p_{\xi}^{2} - mk - mE\xi = -\beta </math>

<math>

2\eta p_{\eta}^{2} - mk - mE\eta = \beta </math>

where β is a constant of motion. Subtraction and re-expression in terms of the Cartesian momenta p_x and P_y shows that β is equivalent to the Laplace-Runge-Lenz vector

<math>

\beta = p_{y} \left( x p_{y} - y p_{x} \right) - mk\frac{x}{r} = A_{x} </math>

This Hamilton-Jacobi approach can be used to derive a conserved generalized Laplace-Runge-Lenz vector <math>\mathcal{A}</math> in the presence of an electric field E <ref name="landau_lifshitz_1976" /><ref>Redmond, PJ (1964). "Generalization of the Runge-Lenz Vector in the Presence of an Electric Field". Physical Review 133: B1352–B1353.</ref>

<math>

\mathcal{A} = \mathbf{A} + \frac{mq}{2} \left[ \left( \mathbf{r} \times \mathbf{E} \right) \times \mathbf{r} \right] </math>

where q is the charge on the orbiting particle.

[edit] Noether's theorem

Noether's theorem states that an infinitesimal variation of the generalized coordinates of a physical system

<math>

\delta q_{i} = \epsilon g_{i}(\mathbf{q}, \mathbf{\dot{q}}, t) </math>

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

<math>

\delta L = \epsilon \frac{d}{dt} G(\mathbf{q}, t) </math>

corresponds to a conserved quantity

<math>

J = -G + \sum_{i} g_{i} \left( \frac{\partial L}{\partial \dot{q}_{i}}\right) </math>

The conserved Laplace-Runge-Lenz vector component A_s corresponds to the variation in the coordinates<ref>Lévy-Leblond, JM (1971). "Conservation Laws for Gauge-Invariant Lagrangians in Classical Mechanics". American Journal of Physics 39: 502–506.</ref>

<math>

\delta x_{i} = \frac{\epsilon}{2} \left[ 2 p_{i} x_{s} - x_{i} p_{s} - \delta_{is} \left( \mathbf{r} \cdot \mathbf{p} \right) \right] </math>

where i equals 1, 2 and 3, with x_i and p_i being the i^th components of the position and momentum vectors r and p, respectively. As usual, δ_is represents the Kronecker delta. The resulting first-order change in the Lagrangian is

<math>

\delta L = \epsilon mk\frac{d}{dt} \left( \frac{x_{s}}{r} \right) </math>

which yields the conserved component A_s

<math>

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 \mathbf{r} \times \mathbf{p} \right]_{s} - mk \left( \frac{x_{s}}{r} \right) </math>

[edit] Lie transformation

Alternatively, the Laplace-Runge-Lenz vector can be derived from a coordinate variation that does not involve the velocities<ref name="prince_eliezer_1981" >Prince, GE, Eliezer CJ (1981). "On the Lie symmetries of the classical Kepler problem". Journal of Physics A: Mathematical and General 14: 587–596.</ref>, by scaling the coordinates r and the time t by different powers of a parameter λ (Figure 6)

<math>

t \rightarrow \lambda^{3}t, \ \mathbf{r} \rightarrow \lambda^{2}\mathbf{r}, \ \mathbf{p} \rightarrow \frac{1}{\lambda}\mathbf{p} </math>

This transformation changes the total angular momentum L and energy E

<math>

L \rightarrow \lambda L, \ E \rightarrow \frac{1}{\lambda^{2}} E </math> but preserves the product EL². Therefore, the eccentricity e and the magnitude A are preserved by the above equation

<math>

A^2 = m^2 k^2 e^{2} = m^2 k^2 + 2 m E L^2 </math>

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 semiaxis a and the period T form a constant. T²/a³.

[edit] Conservation and symmetry

The coordinate variation that leads to the conservation of the Laplace-Runge-Lenz vector by Noether's theorem can be considered as a 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" the electronic orbitals without changing the overall energy. 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 Laplace-Runge-Lenz 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".<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 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 is SO(4), which preserves the length of four-dimensional vectors

<math>

\left| \mathbf{e} \right|^{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 four-dimensional hypersphere.<ref name="fock_1935" /> Specifically, Fock showed that the Schrödinger wavefunction in momentum space for the Kepler problem was the stereographic projection of the spherical harmonics on the hypersphere. Rotation of the hypersphere 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 Laplace-Runge-Lenz vector D formed the Lie algebra for SO(4).<ref name="bargmann_1936" /> 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 four-dimensional hypersphere; 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 four-dimensional hypersphere.

For positive energies — i.e., for unbound, "scattered" systems — the higher symmetry is SO(3,1), which preserves the Minkowski length of 4-vectors

<math>

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">Bander, M, Itzykson C (1966). "Group Theory and the Hydrogen Atom (I)". Reviews of Modern Physics 38: 330–345.</ref><ref>Bander, M, Itzykson C (1966). "Group Theory and the Hydrogen Atom (II)". Reviews of Modern Physics 38: 346–358.</ref>

[edit] Rotational symmetry in four dimensions

The connection between the Kepler problem and four-dimensional rotational symmetry SO(4) can be readily visualized.<ref name="bander_itzykson_1966" /><ref name="rogers_1973">Rogers, HH (1973). "Symmetry transformations of the clasical Kepler problem". Journal of Mathematical Physics 14: 1125–1129.</ref><ref>Guillemin, V, Sternberg S (1990). Variations on a Theme by Kepler. American Mathematical Society Colloquium Publications, volume 42. ISBN 0-8218-1042-1.</ref> 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 <math>\boldsymbol\eta</math> on a four-dimensional unit sphere

<math>

\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} = \frac{mk - r p_{0}^{2}}{mk} \mathbf{\hat{w}} + \frac{rp_{0}}{mk} \mathbf{p} </math>

where <math>\mathbf{\hat{w}}</math> is the unit vector along the new w-axis. Since <math>\boldsymbol\eta</math> has only three independent components, it can be uniquely inverted to obtain p; for example, the x-component equals

<math>

p_{x} = p_{0} \frac{\eta_{x}}{1 - \eta_{w}} </math>

and similarly for p_y and p_z. In other words, the three-dimensional vector p is a stereographic projection of the four-dimensional <math>\boldsymbol\eta</math> vector, scaled by p₀ (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, p_z = η_z = 0; hence, attention may be restricted to the three-dimensional vector <math>\boldsymbol\eta</math> = (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 <math>\boldsymbol\eta</math> sphere, all of which intersect the η_x axis at the two foci η_x = ±1, corresponding to the momentum hodograph foci at p_x = ±p₀. These great circles are related by a simple rotation about that axis (Figure 8). This rotational symmetry transforms all the orbits of the same energy into one another, and is orthogonal to the usual three-dimensional rotational symmetry of central forces, since it transforms the fourth dimension η_w. This higher symmetry is characteristic of the Kepler problem and corresponds to the conservation of the Laplace-Runge-Lenz 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 (α, β, φ)<ref>Lakshmanan, M, Hasegawa H. "On the canonical equivalence of the Kepler problem in coordinate and momentum spaces". Journal of Physics A 17: L889–L893.</ref>

<math>

\eta_{w} = \mathrm{cn}\, \alpha \ \mathrm{cn}\, \beta </math>

<math>

\eta_{x} = \mathrm{sn}\, \alpha \ \mathrm{dn}\, \beta \ \cos \phi </math>

<math>

\eta_{y} = \mathrm{sn}\, \alpha \ \mathrm{dn}\, \beta \ \sin \phi </math>

<math>

\eta_{z} = \mathrm{dn}\, \alpha \ \mathrm{sn}\, \beta </math>

where sn, cn and dn are Jacobi's elliptic functions.

[edit] Generalizations to other potentials and relativity

The Laplace-Runge-Lenz vector has been genernalized to other potentials and even to special relativity. The most general form can be written as<ref name="fradkin_1967" />

<math>

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

<math>

\theta = L \int^{u} \frac{du}{\sqrt{m^{2} c^{2} \left(\gamma^{2} - 1 \right) - L^{2} u^{2}}} </math>

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

<math>

\mathcal{B} = \mathbf{L} \times \mathcal{A} </math>

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

<math>

\mathcal{W} = \alpha \mathcal{A} \otimes \mathcal{A} + \beta \, \mathcal{B} \otimes \mathcal{B} </math>

For illustration, the Laplace-Runge-Lenz vector for a non-relativistic, isotopic harmonic oscillator can be calculated.<ref name="fradkin_1967" /> Since the force is central

<math>

\mathbf{F}(r)= -k \mathbf{r} </math>

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

<math>

\mathbf{W} = \frac{1}{2m} \mathbf{p} \otimes \mathbf{p} + \frac{k}{2} \, \mathbf{r} \otimes \mathbf{r} </math>

although it should be noted that p and r are not perpendicular as A and B are. The corresponding Runge-Lenz vector is more complicated

<math>

\mathbf{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>\omega_{0} = \sqrt{\frac{k}{m}}</math> is the natural oscillation frequency.

[edit] See also

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

[edit] References

[edit] Further reading

• Baez, John. Mysteries of the gravitational 2-body problem.
• Leach, P.G.L., G.P. Flessas (2003). "Generalisations of the Laplace-Runge-Lenz vector". J. Nonlinear Math. Phys. 10: 340–423. arXiv:math-ph/0403028.de:Laplace-Runge-Lenz-Vektor