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

@article{Ballesteros2008HamiltonianSA, title={Hamiltonian Systems Admitting a Runge–Lenz Vector and an Optimal Extension of Bertrand’s Theorem to Curved Manifolds}, author={{\'A}ngel Ballesteros and Alberto Enciso and Francisco J. Herranz and Orlando Ragnisco}, journal={Communications in Mathematical Physics}, year={2008}, volume={290}, pages={1033-1049} }

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 either a harmonic oscillator or a Kepler system. In this paper we extend this classical result to curved spaces by proving that any Hamiltonian on a spherically symmetric Riemannian 3-manifold which satisfies the same conditions as in Bertrand’s theorem is superintegrable and given by an intrinsic…

## 46 Citations

### The classical Darboux III oscillator: factorization, Spectrum Generating Algebra and solution to the equations of motion

- Mathematics
- 2016

In a recent paper the so-called Spectrum Generating Algebra (SGA) technique has been applied to the N-dimensional Taub-NUT system, a maximally superintegrable Hamiltonian system which can be…

### How the modified Bertrand theorem explains regularities of the periodic table I. From conformal invariance to Hopf mapping

- Physics
- 2019

Bertrand theorem permits closed orbits in 3d Euclidean space only for 2 types of central potentials. These are of KeplerCoulomb and harmonic oscillator type. Volker Perlick recently extended Bertrand…

### Classical and quantum higher order superintegrable systems from coalgebra symmetry

- Mathematics, Physics
- 2013

The N-dimensional generalization of Bertrand spaces as families of maximally superintegrable (M.S.) systems on spaces with a nonconstant curvature is analyzed. Considering the classification of…

### Curvature as an Integrable Deformation

- MathematicsIntegrability, Supersymmetry and Coherent States
- 2019

The generalization of (super)integrable Euclidean classical Hamiltonian systems to the two-dimensional sphere and the hyperbolic space by preserving their (super)integrability properties is reviewed.…

### Superintegrable systems on 3-dimensional curved spaces: Eisenhart formalism and separability

- Mathematics
- 2017

The Eisenhart geometric formalism, which transforms an Euclidean natural Hamiltonian H = T + V into a geodesic Hamiltonian T with one additional degree of freedom, is applied to the four families of…

### A generalization of Bertrand's theorem to surfaces of revolution

- Mathematics
- 2011

We prove a generalization of Bertrand's theorem to the case of abstract surfaces of revolution that have no 'equators'. We prove a criterion for exactly two central potentials to exist on this type…

### How the modified Bertrand theorem explains regularities of the periodic table I. From conformal invariance to Hopf mapping

- Physics
- 2019

Bertrand theorem permits closed orbits in 3d Euclidean space only for 2 types of central potentials. These are of Kepler-Coulomb and harmonic oscillator type. Volker Perlick recently extended…

### How the Modified Bertrand Theorem Explains Regularities and Anomalies of the Periodic Table of Elements

- Physics
- 2020

Bertrand theorem permits closed orbits in 3d Euclidean space only for 2 types of central potentials. These are of KeplerCoulomb and harmonic oscillator type. Volker Perlick recently extended Bertrand…

## References

SHOWING 1-10 OF 58 REFERENCES

### Bertrand spacetimes as Kepler/oscillator potentials

- Physics, Mathematics
- 2008

Perlick's classification of (3 + 1)-dimensional spherically symmetric and static spacetimes for which the classical Bertrand theorem holds (Perlick V 1992 Class. Quantum Grav. 9 1009) is revisited.…

### Universal integrals for superintegrable systems on N-dimensional spaces of constant curvature

- Mathematics
- 2006

An infinite family of classical superintegrable Hamiltonians defined on the N-dimensional spherical, Euclidean and hyperbolic spaces are shown to have a common set of (2N − 3) functionally…

### Superintegrable Hamiltonian Systems: Geometry and Perturbations

- Mathematics
- 2005

Abstract
Many and important integrable Hamiltonian systems are ‘superintegrable’, in the sense that there is an open subset of their 2d-dimensional phase space in which all motions are linear on tori…

### Maximal superintegrability on N-dimensional curved spaces

- Mathematics
- 2003

A unified algebraic construction of the classical Smorodinsky–Winternitz systems on the ND sphere, Euclidean and hyperbolic spaces through the Lie groups SO(N + 1), ISO(N) and SO(N, 1) is presented.…

### Superintegrability in a two-dimensional space of nonconstant curvature

- Mathematics
- 2002

A Hamiltonian with two degrees of freedom is said to be superintegrable if it admits three functionally independent integrals of the motion. This property has been extensively studied in the case of…

### Generalized Liouville method of integration of Hamiltonian systems

- Mathematics
- 1978

In this paper we shall show that the equations of motion of a solid, and also Liouville's method of integration of Hamiltonian systems, appear in a natural manner when we study the geometry of level…

### Multifold Kepler systems—Dynamical systems all of whose bounded trajectories are closed

- Mathematics, Physics
- 1995

According to the Bertrand theorem, the Kepler problem and the harmonic oscillator are the only central force dynamical systems that have closed orbits for all bounded motions. In this article, an…

### Superintegrable systems in Darboux spaces

- Mathematics
- 2003

Almost all research on superintegrable potentials concerns spaces of constant curvature. In this paper we find by exhaustive calculation, all superintegrable potentials in the four Darboux spaces of…

### Two classes of dynamical systems all of whose bounded trajectories are closed

- Physics, Mathematics
- 1994

According to the Bertrand theorem, the Kepler problem and the harmonic oscillator are the only central force dynamical systems that have closed orbits for all bounded motions. In this article, other…

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

- Physics
- 1990

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…