Runge-Lenz Research Articles

Generalization of Coulomb classical Runge-Lenz vector to central field model of molecules CH4, SiH4, etc

In the so-called one-centre model of molecules CH4, the protons are smeared uniformly over the surface of a sphere of radiusR equal to the C—H bond length. Here the classical Runge-Lenz vector for the Kepler problem is generalized to this model. Our result reduces to the usual one in the united atom limitR → 0.

Derivation and quantisation of Solov'ev's constant for the diamagnetic Kepler motion

The authors report that the hyperbolic form of the Runge-Lenz vector A, i.e. Lambda (A)=4A2-5Az2, which was shown to be an approximate constant of the diamagnetic Kepler motion by Solov'ev (1982), can be deduced as the lowest non-trivial Birkhoff-Gustavson normal form for a resonant set of generally four oscillators subject to a constraint. Its special case pphi (=m)=0 (i.e. the case of vanishing magnetic quantum number) is shown to agree with the result of Robnik and Schrufer (1985) derived from two oscillators. A systematic scheme of the semiclassical quantisation (the torus quantisation) is discussed on the explicit construction of the genus-one topology, by means of which all the possible, equivalent quantisation formulae are deduced.

Interdimensional degeneracy and symmetry breaking in D-dimensional H+2

An interdimensional degeneracy linking the orbital angular momentum projection ‖m‖ and spatial dimension D gives D-dimensional eigenstates for H+2 by simple correspondence with suitably scaled D=3 excited states. The wave equation for fixed nuclei is separable in D-dimensional spheroidal coordinates, giving generalized two-center differential equations with parametric dependence on the internuclear distance R. By incorporating‖m‖ into D, the resulting eigenstates can be classified by the two dimension-independent ‘‘radial’’ quantum numbers denoted in united atom notation by k and l−‖m‖, corresponding, respectively, to the number of ellipsoidal and hyperboloidal nodal surfaces in the wave function. The two eigenparameters, the energy ED(R), and a separation constant AD(R) related to the total orbital angular momentum and the Runge–Lenz vector, have been determined numerically for the ground state and several low lying excited states for selected dimensions from D=2 to D=100. The system simplifies greatly in the limit D→∞, where the electronic structure reduces to a classical electrostatic form with the electrons in a fixed geometrical configuration relative to the nuclei, akin to the traditional Lewis electron-dot structure. For a given R, the energy E∞ reduces to the minimum of an effective potential surface and the separation constant A∞ reduces to a simple function of the energy. The surfaces are separable in spheroidal coordinates resulting in analytical expressions for the energy in terms of the coordinates. The surfaces exhibit a characteristic symmetry breaking as functions of R, changing from a single minimum surface in the united atom limit (R→0) to a double minimum surface in the separated atom limit (R→∞). Effects of this symmetry breaking are found at finite D as well. Analysis of excited state D-dimensional energies reveals that bonding in H+2 is determined primarily by k, contrary to the standard scheme of bonding and antibonding molecular orbitals, which in the case of H+2 correspond to even and odd l−‖m‖, respectively. When the D-dimensional energies are examined as functions of 1/D, the resulting curves resemble typical perturbation diagrams with 1/D as the perturbation parameter.

Right and wrong use of the Lenz vector for non-Newtonian potentials

An article that recently appeared in this Journal1 propagates an error first published here more than 20 years ago.2 This concerns a proposed use of the Lenz vector3 in obtaining the rate of precession of the perihelion for a central-force problem in which the potential has a small departure from the Newtonian form, V(r)=k/r. Three correct solutions to this problem are presented, based on the effective-potential technique, on the method of averages, and on a careful use of the Lenz vector.4

A generalisation of the Runge-Lenz constant of classical motion in a central potential

A perihelion (unit) vector-the generalisation of the Runge-Lenz vector obtained by Fradkin (1967) is investigated. Its evolution, viewed as its dependence on the position and momentum vectors of a particle moving along its trajectory, demonstrates that, in general, it depends on time. For trajectories having the form of closed orbits with one pair of turning points, the perihelion vector turns out to be time independent and therefore a true integral of the motion. When a closed orbit possesses n pairs of turning points, an n-arm star of n different perihelion vectors is an invariant of the motion. In this case integrals of motion in the form of an n-rank tensor can be constructed from the perihelion vectors. Conditions for the existence of closed orbits are derived and illustrated by examples of the motion in the Kepler potential perturbed by a centrifugal-like term and in the isotropic harmonic oscillator potential similarly perturbed.

Calculation of H2+ eigenparameters in arbitrary dimensions

Abstract An interdimensional degeneracy linking the orbital angular momentum projection |m| and spatial dimension D yields D-dimensional eigenstates for H2+ by simple correspondence with suitably scaled D = 3 excited states. The wave-equation for fixed nuclei is separable in D-dimensional spheroidal coordinates, giving generalized two-center differential equations with parametric dependence on the internuclear distance R. A computer program is presented which is capable of calculating the two eigenparameters, the energy ED(R), and the separation constant AD(R) related to the total orbital angular momentum and the Runge-Lenz vector, to high accuracy over a large range in R for dimensions up to D = 100, or equivalently, for projections up to |m| = 50 for D = 3. The program can be implemented on personal computers to give accuracies up to 16 digits. The execution times required are modest. Normalization constants, wavefunctions and expectation values can be calculated as well, but at a cost of much larger execution times for even moderate accuracy.

Wrong-way recursion yields more accurate eigenparameters for the hydrogen-molecule ion.

In the power-series solution of the separated two-center equations for ${\mathrm{H}}_{2}^{+}$, determining eigenparameters by recursion in the stable direction (from high to low order) encounters severe numerical limitations in the case of large internuclear distance and high spatial dimension or, equivalently, high angular momentum. We find this can be overcome by use of recursion in the unstable direction (from low to high order), thereby greatly improving the accuracy of the eigenparameters obtained. For instance, for the ground electronic state with angular momentum \ensuremath{\Vert}m\ensuremath{\Vert}=0 and dimension D=67 (equivalent to \ensuremath{\Vert}m\ensuremath{\Vert}=32 for D=3), the improvement in accuracy is about five orders of magnitude for the energy and about eight orders for the separation constant related to the Runge-Lenz vector. The source of this seemingly paradoxical behavior is revealed by examining the divergent solution for each recursion direction. The method is useful for determining the eigenparameters only; recursion in the stable direction must still be used to determine the eigenfunctions.

Non-relativistic Coulomb problem in a one-dimensional quantum mechanics

It is shown that, unlike the model based only on the potential singularity U(x)=- mod x mod -1, the group of hidden symmetry explains not only the double degeneration of the energy spectrum but also the explicit form of the spectrum, wavefunctions and an extra constant of motion analogous to the Runge-Lenz vector.

Quantum wave packets on Kepler elliptic orbits.

Wave-packet solutions of the Schr\"odinger equation for the Coulomb potential are obtained that travel along classical elliptic orbits of fixed mean eccentricity and angular momentum. These wave packets are coherent states that have minimal quantum fluctuations in the noncommuting components of the Lenz vector in the plane of the orbit. For large quantum numbers the asymptotic form of the wave packet and the classical equations of the orbit are obtained analytically.

Laplace vectors for the harmonic oscillator

The two invariant Laplace vectors for the two-dimensional harmonic oscillator are derived from an elementary point of view. They are used to determine the equation of the orbit, and the orientation of the orbit from the initial conditions of the motion.

Conserved vectors and orbit equations for autonomous systems governed by the equation of motion r̈+f ṙ+g r=0

The existence of conserved vectors that are analogous to the conserved angular momentum and Laplace–Runge–Lenz vectors of the classical Kepler problem makes it possible to determine the orbit equations for a wide variety of autonomous systems governed by the equation of motion of the form r̈+f ṙ+gr=0.

Determination of the generalized Laplace–Runge–Lenz vector by an inverse matrix method

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Winding strings, Kaluza-Klein monopoles and runge-lenz vectors

We study the interaction of a class of closed solitonic string states (“winders”) which wind around the Kaluza-Klein circle with a Kaluza-Klein monopole. We find that winders are attracted towards monopoles with an inverse-square law force and even when moving relativistically they pursue exactly elliptical parabolic or hyperbolic orbits. The simplicity of the motion is due to the existence of an extra conserved Runge-Lenz vector in addition to the conserved angular momentum. The Poisson algebra of these conserved vectors closes on so (3) ⊕ s R 3 or so (3), so (3) ⊕ s R or so (3) ⊕ so (2, 1). We also find a class of instanton solutions of the euclidean equations in the field of two or more monopoles. The possibility that monopoles catalyse the decay or annihilation of winders is discussed.

Equations of motion using the dynamical evolution of the Runge-Lenz vector

view Abstract Citations (4) References (7) Co-Reads Similar Papers Volume Content Graphics Metrics Export Citation NASA/ADS Equations of Motion Using the Dynamical Evolution of the Runge-Lenz Vector Bartnik, E. A. ; Haberzettl, H. ; Sandhas, W. Abstract Based on the dynamical evolution of the Runge-Lenz vector, a set of first-order differential equations of motion for the calculation of orbits for arbitrary forces is given. The corresponding orbit equation is explicitly expressed via the time-dependent Runge-Lenz vector ɛ as a local conic section relative to ɛ, with the local eccentricity given by |ɛ|. While completely general, this approach is particularly well suited for those problems in celestial mechanics which can be formulated as pertubations of the Kepler case. As an example, the authors treat the motion of an asteroid in the gravitational force fields of the Sun and Jupiter within framework of the classical restricted three-body problem. Publication: The Astrophysical Journal Pub Date: November 1988 DOI: 10.1086/166856 Bibcode: 1988ApJ...334..517B Keywords: Asteroids; Celestial Mechanics; Dynamic Characteristics; Equations Of Motion; Planetary Orbits; Vector Analysis; Computational Astrophysics; Gravitational Fields; Three Body Problem; Time Dependence; Astrophysics; ASTEROIDS; CELESTIAL MECHANICS full text sources ADS |

On a hidden symmetry of a relativistic Coulomb problem in the quasipotential approach

The Runge-Lenz vector for a relativistic Coulomb problem in the quasipotential approach is found. The energy spectrum is obtained by an algebraic method. A hidden symmetry group of this problem is shown to be a group O(4) for mod E mod 1 and a motion group of the three-dimensional Euclidean space for mod E mod =1.

Considerations on the precessing orbit via a rotating Laplace–Runge–Lenz vector

For the relativistic Kepler–Coulomb problem, a rotating Laplace–Runge–Lenz vector is constructed as an extension of the well-known conserved vector for the Kepler–Coulomb problem. Motion of the turning point in the precessing orbit is analyzed from the considerations on the coordinate systems related to the rotating vector.

The Rutherford cross section and the perihelion shift of Mercury with the Runge–Lenz vector

It is shown how the Runge–Lenz vector A can be used to give the orbit equation and the Rutherford cross section without the necessity of solving differential equations. After that, the Greenberg’s method is applied to obtain the perihelion shift of Mercury using the vector A. Some suggestions are given in the conclusions.

The effect of a combined static and microwave field on an excited hydrogen atom

The authors examine the behaviour of extremal Stark states of excited hydrogen under the influence of parallel static and microwave electric fields in order to assess when the system can be approximated by a Hamiltonian with one degree of freedom. The authors show that without a static field the Runge-Lenz vector precesses, so that only for short times does the electron remain approximately aligned along the field direction. In this case they provide estimates of the time during which the motion is nearly one-dimensional; this has a complicated dependence upon the microwave frequency. The amplitude of the Runge-Lenz oscillations decreases with increasing static field strength Fs; there is a critical static field, depending upon the microwave field frequency Omega and amplitude Fm, at which these oscillations are small enough for the electron motion to be considered one-dimensional. The authors provide estimates of this critical field for various microwave frequencies and show that it is largest when the microwave field is in resonance with the electron motion, in which case it is necessary that Fs0.22Fm.

Coulomb–Kepler problem and the harmonic oscillator

The three-dimensional Coulomb–Kepler problem is shown to be equivalent to a pair of coupled two-dimensional harmonic oscillators with the same angular momentum in classical mechanics by means of a Kustaanheimo–Stiefel transformation of coordinates and velocities. The constraint condition inherent in the transformation is shown to be related to the Runge–Lenz vector. The relationship between the action variables of the two systems is discussed. The equivalence is seen to result in the separability of the problem in parabolic coordinates.

The Kepler problem recast: Use of a transverse velocity transformation and the invariant velocities

For conservative central forces the constancy of angular momentum provides a more natural substitution than is normally used to transform the force law into a differential equation. This is a transverse velocity substitution, which emphasizes angular rather than radial dependence for dynamical variables, implying a functional rotation. The method gives the standard results in new forms and they are expressed in more natural units of length, time, velocity, and energy than appears in other approaches. For the inverse square force the transverse velocity explicitly depends on the two invariant velocities, and the total energy is the difference of the kinetic energies for the two velocities. The invariant angular momentum is the sum of two variable angular momenta for the two velocities. The Laplace–Runge–Lenz vector is one of a triplet of orthogonal invariant vectors, the cross product of the invariant momentum and angular momentum. For noncircular trajectories this triplet and the scalar total energy are constants of the motion.