# Eikonal Blog

## 2010.10.08

### Runge-Lenz vector

For a potential $U(\vec{r}) = \frac{k}{r}$, the conserved vectors are (slightly changed definitions from reference [1]):

• the angular momentum verctor: $\vec{L} :\equiv \vec{r}\times\vec{p}$, and
• the (dimensionless) eccentricity vector: $\vec{e} = \frac{\underline{A}}{mk} :\equiv \frac1{mk}\vec{p}\times\vec{L} - \hat{r}$ where $\hat{r} :\equiv \frac1{r} \vec{r}$, and $\underline{A}$ is the LRL vector (the Laplace-Runge-Lenz vector = the Runge–Lenz vector = the Lenz vector).
• As a consequence, the (dimensionless) vector of binormal is also preserved: $\vec{b} :\equiv \vec{l}\times\vec{e} = \frac{L}{mk}\vec{p} +\hat{r}\times\vec{l}$. Here $\vec{l}:\equiv \frac1{L}\vec{L} = \vec{e}\times\vec{b}$.

• [1] Wikipedia: Laplace-Runge-Lenz vector – http://en.wikipedia.org/wiki/Laplace%E2%80%93Runge%E2%80%93Lenz_vector | Eccentricity vector – http://en.wikipedia.org/wiki/Eccentricity_vector
• Laplace-Runge-Lenz Vector – http://scienceworld.wolfram.com/physics/Laplace-Runge-LenzVector.html
• Laplace-Runge-Lenz vector (The Tangent Bundle Physics Wiki) – http://www.physics.thetangentbundle.net/wiki/Quantum_mechanics/Laplace-Runge-Lenz_vector
• P.E.S. Wormer (2003). Properties of the quantum mechanical Runge-Lenz vector. – <a href=http://www.theochem.ru.nl/~pwormer/rungelenz.pdf".http://www.theochem.ru.nl/~pwormer/rungelenz.pdf
• “Mysteries of the gravitational 2-body problem” by John Baez (2003.05.03) – http://math.ucr.edu/home/baez/gravitational.html
• “A planar Runge-Lenz vector” by S.G.Kamath (arXiv:hep-th/0112067v1; 2001.12.10; also J.Math.Phys. 43 (2002) 318-324) – http://arxiv.org/abs/hep-th/0112067
Abstract: Following Dahl’s method an exact Runge-Lenz vector M with two components M and M is obtained as a constant of motion for a two particle-system with charges e and e whose electromagnetic interaction is based on Chern-Simons electrodynamics. The Poisson bracket {M, M} = L but is modified by the appearance of the product e e as central charges.
• “Duality of force laws and Conformal transformations” by Dawood Kothawala
(arXiv:1010.2238v3 [physics.class-ph]; 2011.01.07) – http://arxiv.org/abs/1010.2238

Abstract: As was first noted by Isaac Newton, the two most famous ellipses of classical mechanics, arising out of the force laws F~r and F~1/r^2, can be mapped onto each other by changing the location of center-of-force. What is perhaps less well known is that this mapping can also be achieved by the complex transformation, z -> z^2. We give a simple derivation of this result (and its generalization) by writing the Gaussian curvature in its “covariant” form, and then changing the \emph{metric} by a conformal transformation which “mimics” this mapping of the curves. The final result also yields a relationship between Newton’s constant G, mass M of the central attracting body in Newton’s law, the energy E of the Hooke’s law orbit, and the angular momenta of the two orbits. We also indicate how the conserved Laplace-Runge-Lenz vector for the 1/r^2 force law transforms under this transformation, and compare it with the corresponding quantities for the linear force law. Our main aim is to present this duality in a geometric fashion, by introducing elementary notions from differential geometry.
• “Laplace-Runge-Lenz symmetry in general rotationally symmetric systems” by Uri Ben-Ya’acov (arXiv:1005.1817; 2010.05.11) – http://arxiv.org/abs/1005.1817
Abstract: The Laplace-Runge-Lenz symmetry, well known to exist in classical two-body Kepler-Coulomb systems, is also known to be generalizable to all rotationally symmetric systems. It also appears in the computation of the Lorentz boost in relativistic systems. Towards the unification of these properties, the generic (independent of the interaction) properties of the symmetry are verified and extended. The independence of the symmetry on the type of interaction is proven applying only the most minimal properties of the Poisson brackets. Generalized Laplace-Runge-Lenz vectors are definable to be constant (not only piece-wise conserved) for all cases, including open orbits. Also discussed are the transformations generated by the Laplace-Runge-Lenz vectors, the emergence of these vectors in post-Newtonian extensions of general centrally symmetric systems, and the application of these results to relativistic Coulomb systems.
• “Generalized Laplace-Runge-Lenz vector for the three-dimensional classical motions generated by central forces with a monopole” by T. Yoshida (Il Nuovo Cimento B (1971-1996); Volume 104, Number 4, 375-385, DOI: 10.1007/BF02725670) – http://www.springerlink.com/content/k481322u42ln7244/ [REQUIRES PAYMENT FOR ACCESS]
Abstract: This paper is concerned with the general method of constructing conserved vectors, which is applicable to noncentral force problems. The point of this method is to express an orthonormal system of vectors in terms of the position, momentum and angular-momentum vectors. Then conserved vectors, such as the Laplace-Runge-Lenz vector, are obtained as a unit vector of the fixed orthonormal system, if the equations of motion are solvable by the quadrature. To facilitate the procedure for obtaining the conserved vectors a unit dyadic is introduced. A generalized Laplace-Runge-Lenz vector is obtained indeed by this procedure for the three-dimensional classical motions generated by central forces with Dirac’s monopole.
• “Energy spectrum of the two-dimensionalq-hydrogen atom” by Shengli Zhang (International Journal of Theoretical Physics; Volume 34, Number 11, 2217-2221, DOI: 10.1007/BF00673837) – http://www.springerlink.com/content/n251256l88572074/ [REQUIRES PAYMENT FOR ACCESS]
Abstract: The discrete energy spectrum of theq-analog of the two-dimensional hydrogen atom is derived by deforming the Pauli equation. It contracts to that of the ordinary two-dimensional hydrogen atom in the limitq rarr ± 1. The degeneracy is discussed generally and some properties of theq-energy spectrum are studied both for realq and for complexq of magnitude unity.
• “The hidden symmetry of the Coulomb problem in relativistic quantum mechanics: From Pauli to Dirac” by Tamari T. Khachidze and Anzor A. Khelashvili (American Journal of Physics — July 2006 — Volume 74, Issue 7, pp. 628-632) – http://ajp.aapt.org/resource/1/ajpias/v74/i7/p628_s1 [REQUIRES PAYMENT FOR ACCESS]
Abstract: Additional conserved quantities associated with an extra symmetry govern a wide variety of physical systems ranging from planetary motion to atomic spectra. We give a simple derivation of the hidden symmetry operator for the Dirac equation in a Coulomb field and show that this operator may be reduced to the one introduced by Johnson and Lippmann to include the spin degrees of freedom. This operator has been rarely discussed in the literature and has been neglected in recent textbooks on relativistic quantum mechanics and quantum electrodynamics.
• “Rotating Laplace-Runge-Lenz vector leading to two relativistic Kepler’s equations” by Takeshi Yoshida (Phys. Rev. A 38, 19–25 (1988) = Phys. Rev. A » Volume 38 » Issue 1) – http://pra.aps.org/abstract/PRA/v38/i1/p19_1 [just abstract]
Abstract: For the nonrelativistic Kepler problem it is well known that there exists a transcendental equation (the so-called Kepler’s equation) which gives a position in the orbit at a given time. The precessing orbit of the relativistic Kepler problem is reduced to the nonprecessing one by using the transformation related to a rotating Laplace-Runge-Lenz vector. This vector is an extension of the familiar conserved vector and is always oriented to the moving perihelion point of the precessing orbit. From the considerations on Kepler’s equation and the rotating Laplace-Runge-Lenz vector, a relativistic Kepler’s equation is newly defined to find a position in the precessing orbit at a given time. This equation has two expressions corresponding to the relativistic coordinate time and proper time. By using the equation, Lambert’s theorem that gives the required time on the orbit is extended to the relativistic problem.
• “Dynamics of the Laplace-Runge-Lenz vector in the quantum-corrected Newton gravity” by C. Farina, W. J. M. Kort-Kamp, Sebastiao Mauro Filho, Ilya L. Shapiro (arXiv:1101.5611v2 [gr-qc]; 2011.02.02) – http://arxiv.org/abs/1101.5611
Abstract: Recently it was shown that quantum corrections to the Newton potential can explain the rotation curves in spiral galaxies without introducing the Dark Matter halo. The unique phenomenological parameter $\al\nu$ of the theory grows with the mass of the galaxy. In order to better investigate the mass-dependence of $\al\nu$ one needs to check the upper bound for $\al\nu$ at a smaller scale. Here we perform the corresponding calculation by analyzing the dynamics of the Laplace-Runge-Lenz vector. The resulting limitation on quantum corrections is quite severe, suggesting a strong mass-dependence of $\al\nu$.
• “Orbit Determination with the two-body Integrals” by Giovanni Federico Gronchi, Linda Dimare, Andrea Milani (arXiv:0911.3555v2 [math-ph]; 2010.03.31) – http://arxiv.org/abs/0911.3555
Abstract: We investigate a method to compute a finite set of preliminary orbits for solar system bodies using the first integrals of the Kepler problem. This method is thought for the applications to the modern sets of astrometric observations, where often the information contained in the observations allows only to compute, by interpolation, two angular positions of the observed body and their time derivatives at a given epoch; we call this set of data attributable. Given two attributables of the same body at two different epochs we can use the energy and angular momentum integrals of the two-body problem to write a system of polynomial equations for the topocentric distance and the radial velocity at the two epochs. We define two different algorithms for the computation of the solutions, based on different ways to perform elimination of variables and obtain a univariate polynomial. Moreover we use the redundancy of the data to test the hypothesis that two attributables belong to the same body (linkage problem). It is also possible to compute a covariance matrix, describing the uncertainty of the preliminary orbits which results from the observation error statistics. The performance of this method has been investigated by using a large set of simulated observations of the Pan-STARRS project.
• “Orbit Determination with the two-body Integrals. II” by Giovanni F. Gronchi, Davide Farnocchia, Linda Dimare (arXiv:1101.4569v1 [math-ph]; 2011.01.24) – http://arxiv.org/abs/1101.4569
Abstract: The first integrals of the Kepler problem are used to compute preliminary orbits starting from two short observed arcs of a celestial body, which may be obtained either by optical or radar observations. We write polynomial equations for this problem, that we can solve using the powerful tools of computational Algebra. An algorithm to decide if the linkage of two short arcs is successful, i.e. if they belong to the same observed body, is proposed and tested numerically. In this paper we continue the research started in [Gronchi, Dimare, Milani, ‘Orbit determination with the two-body intergrals’, CMDA (2010) 107/3, 299-318], where the angular momentum and the energy integrals were used. A suitable component of the Laplace-Lenz vector in place of the energy turns out to be convenient, in fact the degree of the resulting system is reduced to less than half.
• “The Universal Kepler Problem” by Guowu Meng (arXiv:1011.6609v3 [math-ph]; 2010.12.22) – http://arxiv.org/abs/1011.6609>
Abstract: For each simple euclidean Jordan algebra, the analogues of hamiltonian, angular momentum and Lenz vector in the Kepler problem are introduced. The analogue of hidden symmetry algebra generated by hamiltonian, angular momentum and Lenz vector is also derived. Finally, for the simplest simple euclidean Jordan algebra, i.e., $\bb R$, we demonstrate how to get generalized Kepler problems by combining with the quantizations of the TKK algebra.
• pages 90-95 of book “Hamiltonian dynamics” by Gaetano Vilasi – http://tinyurl.com/648t4zf
• “Determination of the Runge—Lenz Vector” by W. H. Heintz (1974) – http://www.phys.ufl.edu/~maslov/classmech/heinz.pdf [PDF]