# Generalisations of the Laplace–Runge–Lenz Vector

@article{Leach2003GeneralisationsOT, title={Generalisations of the Laplace–Runge–Lenz Vector}, author={Peter G. L. Leach and George Flessas}, journal={Journal of Nonlinear Mathematical Physics}, year={2003}, volume={10}, pages={340 - 423} }

Abstract The characteristic feature of the Kepler Problem is the existence of the so-called Laplace–Runge–Lenz vector which enables a very simple discussion of the properties of the orbit for the problem. It is found that there are many classes of problems, some closely related to the Kepler Problem and others somewhat remote, which share the possession of a conserved vector which plays a significant rôle in the analysis of these problems.

## 41 Citations

### On the Equivalence Between Manev and Kepler Problems

- Mathematics, Physics
- 2008

Here we demonstrate the existence of a local Darboux chart for the Manev model such that its dynamics becomes locally equivalent to the Kepler model. This explains lot of similarities between these…

### Laplace–Runge–Lenz symmetry in general rotationally symmetric systems

- Physics
- 2010

The universality of the Laplace–Runge–Lenz symmetry in all rotationally symmetric systems is discussed. The independence of the symmetry on the type of interaction is proven using only the most…

### Superintegrable systems on sphere

- Mathematics, Physics
- 2005

We consider various generalizations of the Kepler problem to three-dimensional sphere $S^3$, a compact space of constant curvature. These generalizations include, among other things, addition of a…

### II. COMPLEX DESCRIPTION OF MOTIONS WITH CONSERVATION OF THE DIRECTION OF ANGULAR MOMENTUM

- Physics
- 2008

In the category of motions preserving the angular momentum’s direction, Gorringe and Leach exhibited two classes of differential equations having elliptical orbits. After enlarging slightly these…

### Bohlin-Arnold-Vassiliev's duality and conserved quantities

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- 2008

Bohlin-Arnold-Vassiliev's duality transformation establishes a correspondence between motions in different central potentials. It offers a very direct way to construct the dynamical conserved…

### Hamiltonian Systems Admitting a Runge–Lenz Vector and an Optimal Extension of Bertrand’s Theorem to Curved Manifolds

- Mathematics
- 2008

Bertrand’s theorem asserts that any spherically symmetric natural Hamiltonian system in Euclidean 3-space which possesses stable circular orbits and whose bounded trajectories are all periodic is…

### Complex functions and geometric structures associated to the superintegrable Kepler-related family of systems endowed with generalized Runge-Lenz integrals of motion

- Mathematics
- 2020

The existence of quasi-bi-Hamiltonian structures for a two-dimensional superintegrable $(k_1,k_2,k_3)$-dependent Kepler-related problem is studied. We make use of an approach that is related with the…

### On the symmetry analysis of the micz problem on the cone

- Mathematics
- 2016

In this paper we used reduction method of Arunaye [1] to obtain the symmetries of the McIntosh and Cisneros [2]; and Zwanziger [3] (MICZ) problem and present some exact symmetry transformations of…

### Elementary derivation of the Lense-Thirring precession

- Physics
- 2008

An elementary pedagogical derivation of the Lense-Thirring precession is given based on the use of Hamilton vector. The Hamilton vector is an extra constant of motion of the Kepler/Coulomb problem…

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