Runge-Lenz Research Articles

Algebraic Solution of the Kepler Problem Using the Runge-Lenz Vector

The Runge-Lenz vector is used to obtain the equation for the Kepler orbits algebraically and the frequency of small oscillations of a particle about a stable circular orbit. The invariance of the Hamiltonian under the Lie algebras generated by the components of the Runge-Lenz and angular momentum vectors is discussed.

Physical interpretation of the Runge-Lenz vector

The Runge-Lenz vector, which is a constant of the motion in the nonrelativistic hydrogen atom, is shown to be a two-body quantity, whose presence can be completely understood within the classical framwork of special relativity.

A constant of the motion for the two‐centre Kepler problem

AbstractThe one‐centre Coulombic potential gives rise, in non‐relativistic mechanics, to additional constants of the motion which form the components of the Runge–Lenz vector. By a study of this vector, an extra constant of the motion is derived for the corresponding two‐centre problem. The result holds quite generally in a space of arbitrary dimension and is applicable to both classical and quantum mechanics; but breaks down when relativistic corrections, which destroy the extra symmetry of the one‐centre Coulombic potential, are taken into account. The effect of further Coulombic centres and of varying the form of the potential is briefly discussed. In particular a constant of the motion is derived for a two‐centre potential which has both Coulombic and simple harmonic terms. The relationship between these constants of the motion and the separation of the Hamiltonian into spheroidal coordinates is noted (this had previously been known only for the two‐centre Coulomb problem in three‐dimensional space). Finally the application to the hydrogen molecule ion, treated in the adiabatic approximation, is pointed out. The extra constant of the motion is seen to account for an observed apparent breakdown in the noncrossing rule for the potential energy curves.

Conserved quantities and symmetry groups for the Kepler problem

Generators of the «invariance» dynamical groupsO 4,E 3 andO 3.1 (for bound states) andE 3 andO 3.1 (for scattering states) are explicitly constructed in terms of the dynamical variables. It is shown that two families of generators exist (one of them containing the familiar Runge-Lenz vector), each one of them depending on an arbitrary function of the energy, leading to an ambiguity in the relation of the Hamiltonian with the Casimir operators of the «invariance» dynamical groups.

Existence of the Dynamic SymmetriesO4andSU3for All Classical Central Potential Problems

It is found that all classical dynamic problems (relativistic as well as non-relativistic) involving central potentials, inherently possess both O4 and SU3 symmetry. This leads to a generalization of both the Runge-Lenz vector in the Kepler problem and the conserved symmetric tensor in the harmonic oscillator problem. For a general central potential, an explicit construction of the elements of the Lie algebra of O4 and SU3 in terms of canonical variables is given. The question of a possible quantum-mechanical analog is discussed. Also, a constructive technique is given for imbedding the Lorentz group and SU3 in an infinite-dimensional Lie algebra.

Runge-Lenz Vector and the Coulomb Green's Function

The fact that the Runge-Lenz vector is an extra constant of the motion for a charged particle moving in a Coulomb potential is found to account for the especially simple structure of the nonrelativistic coordinate space Coulomb Green's function. Also, the study of the consequences of this extra constant of the motion leads to a separation of variables in the differential equation for the coordinate space Coulomb Green's function, and hence to a new derivation of the closed-form expression for this Green's function which avoids the use of infinite series and the detailed properties of special functions.

Dynamical groups and spherical potentials in Classical Mechanics

The one particle problem in a spherical potential is examined in Classical Mechanics from a group theorical point of view. The constants of motion are classified according to their behaviour under the rotation groupSO(3), i.e. according to the irreducible representationsD j ofSO(3) (section 1). The Lie algebras ofSO(4) andSO(3) are explicitly built in terms of Poisson brackets for an arbitrary potential, from global considerations. The Kepler and the 3 dimensional oscillator problems are shown to play particular roles with respect to these groups (sections 2 and 3). In the last section, the Kepler problem is analyzed with the aid of theSO(4) group instead of the Lie algebra. It is proved that the transformations generated by the angular momentum and the Runge-Lenz vector form indeed a group of canonical transformations isomorphic toSO(4). Consequences with respect to the quantization problem are examined.

Three-Dimensional Isotropic Harmonic Oscillator and SU3

A consideration of the eigenvalue problem for the quantum-mechanical three-dimensional isotropic harmonic oscillator leads to a derivation of a conserved symmetric tensor operator, in addition to the angular momentum vector operator. Upon examining the classical limit, it is found that the symmetric tensor completely specifies the orientation of the elliptical orbit, in a way analogous to the Runge-Lenz vector for the Kepler problem. The tensor operator and the angular momentum vector operator are then related to the infinitesimal operators for the SU3 group.

Generalization of the Runge-Lenz Vector in the Presence of an Electric Field

It is well known that the Kepler problem admits two vector constants of the motion, the angular momentum and the Runge-Lenz vector. In this paper a generalization of the Runge-Lenz vector is found when a uniform constant electric field is present. The component of this vector in the direction of the external field is a constant of the motion.