Quasi-Bi-Hamiltonian structures of the 2-dimensional Kepler problem

@article{Cariena2016QuasiBiHamiltonianSO,
  title={Quasi-Bi-Hamiltonian structures of the 2-dimensional Kepler problem},
  author={Jos{\'e} F. Cari{\~n}ena and Manuel F Ra{\~n}ada},
  journal={Symmetry Integrability and Geometry-methods and Applications},
  year={2016},
  volume={12},
  pages={010}
}
The existence of quasi-bi-Hamiltonian structures for the Kepler problem is studied. We first relate the superintegrability of the system with the existence of two complex functions endowed with very interesting Poisson bracket properties and then we prove the existence of a quasi-bi-Hamiltonian structure by making use of these two functions. The paper can be considered as divided in two parts. In the first part a quasi-bi-Hamiltonian structure is obtained by making use of polar coordinates and… 
Bi-Hamiltonian structure of the bi-dimensional superintegrable nonlinear isotonic oscillator
The higher-order superintegrability of the two-dimensional isotonic oscillator (noncentral oscillator with inversely quadratic nonlinearities also known as caged anisotropic oscillator) with rational
Hamiltonian Dynamics for the Kepler Problem in a Deformed Phase Space
This work addresses the Hamiltonian dynamics of the Kepler problem in a deformed phase space, by considering the equatorial orbit. The recursion operators are constructed and used to compute the
Quasi-bi-Hamiltonian structures and superintegrability: Study of a Kepler-related family of systems endowed with generalized Runge-Lenz integrals of motion
The existence of quasi-bi-Hamiltonian structures for a two-dimen-sional superintegrable \begin{document}$ (k_1,k_2,k_3) $\end{document} -dependent Kepler-related problem is studied. We make use of an
Complex functions and geometric structures associated to the superintegrable Kepler-related family of systems endowed with generalized Runge-Lenz integrals of motion
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
Quasi-bi-Hamiltonian structures, complex functions and superintegrability: the Tremblay–Turbiner–Winternitz (TTW) and the Post–Winternitz (PW) systems
The existence of quasi-bi-Hamiltonian structures for the Tremblay–Turbiner–Winternitz (TTW) and the Post–Winternitz (PW) systems is studied. We first recall that the superintegrability of these two

References

SHOWING 1-10 OF 37 REFERENCES
Two degrees of freedom quasi-bi-Hamiltonian systems
Starting from the classical example of the Henon - Heiles integrable Hamiltonian system, we show that it admits a slightly different formulation from the classical bi-Hamiltonian system. We introduce
Dynamical symmetries, bi-Hamiltonian structures, and superintegrable n=2 systems
The theory of dynamical but non-Cartan (or non-Noether) symmetries and the existence of bi-Hamiltonian structures is studied using the symplectic formalism approach. The results are applied to the
On bi-Hamiltonian formulation of the perturbed Kepler problem
The perturbed Kepler problem is shown to be a bi-Hamiltonian system in spite of the fact that the graph of the Hamilton function is not a hypersurface of translation, which goes against a necessary
Families of quasi-bi-Hamiltonian systems and separability
It is shown how to construct an infinite number of families of quasi-bi-Hamiltonian (QBH) systems by means of the constrained flows of soliton equations. Three explicit QBH structures are presented
Completely integrable bi-Hamiltonian systems
We study the geometry of completely integrable bi-Hamiltonian systems and, in particular, the existence of a bi-Hamiltonian structure for a completely integrable Hamiltonian system. We show that
A new proof of the higher-order superintegrability of a noncentral oscillator with inversely quadratic nonlinearities
The superintegrability of a rational harmonic oscillator (non-central harmonic oscillator with rational ratio of frequencies) with non-linear "centrifugal" terms is studied. In the first part, the
Quasi-bi-Hamiltonian systems and separability
Two quasi-bi-Hamiltonian systems with three and four degrees of freedom are presented. These systems are shown to be separable in terms of Nijenhuis coordinates. Moreover, the most general Pfaffian
...
...