On bi-Hamiltonian formulation of the perturbed Kepler problem

Abstract

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 is against a necessary condition for the existence of the bi-Hamiltonian structure according to the Fernandes theorem. In fact, both the initial and perturbed Kepler systems are isochronous systems and, therefore, the Fernandes theorem cannot be applied to them.

Quasi-Bi-Hamiltonian Structures of the 2-Dimensional Kepler Problem

“…The potential of the Kepler problem is spherically symmetric and therefore it admits alternative Lagrangians (the existence of alternative Lagrangians for central potentials is studied in [13,20,26]); recall that if there exist alternative Lagrangian descriptions then one can find non-Noether constants of motion [7]. This system has been studied as a bi-Hamilltonian system by making use of different approaches; Rauch-Wojciechowski proved the existence of a bi-Hamiltonian formulation but introducing an extra variable so that the phase space is odddimensional and the Poisson brackets are degenerate [34] and more recently [19] a bi-Hamiltonian formulation for the perturbed Kepler problem has also been studied by making use of Delaunaytype variables.…”

Noncommutative Kepler Dynamics: symmetry groups and bi-Hamiltonian structures

“…In particular, we use here the Hamiltonian function H in the defined Delaunay-type variable (I, φ) to construct bi-Hamiltonian structures. Now, following the example of the generic Bogoyavlenskij construction for the isochronous Hamiltonian system proposed by Grigoryev et al in 2015 [28], we can carry out the following canonical transformations:…”

“…Over the past few years, Magri's approach [41] to integrability through bi-Hamiltonian structures has became one of the most powerful methods relating to the integrability of evolution equations, applicable in studying both finite and infinite dimensional dynamical systems [28]. This approach has also been proven to be one of the classical methods of integrability of evolution equations along with, for example, the Hamilton-Jacobi method of separation of variables and the method of the Lax representation [35,55].…”

“…The application of Smirnov's result to the classical Kepler problem is due to the possibility of transforming the action-angle coordinates connected with the spherical-polar coordinates to the Delaunay coordinates. In 2005, Rañada [51] proved the existence of a bi-Hamiltonian structure arising from a non-symplectic symmetry as well as the existence of master symmetries and additional integrals of motion (weak superintegrability) for certain particular values of two parameters b and k. In 2015, Grigoryev et al showed that the perturbed Kepler problem is 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 condition for the existence of the bi-Hamiltonian structure according to the Fernandes theorem [21,28]. They explicitly presented a few non-degenerate bi-Hamiltonian formulations of the perturbed Kepler problem using the Bogoyavlenskij construction of a continuum of compatible Poisson structures for the isochronous Hamiltonian systems [8].…”

Hamiltonian Dynamics for the Kepler Problem in a Deformed Phase Space

“…Further, in 2013, Hosokawa and Takeuchi [15] solved the same problem, but using the Runge-Lenz-Pauli vector, and got new constants of motion. A bi-Hamiltonian formulation for a Kepler problem was also studied with Delaunay-type variables [14]. In 2016, J. F. Cariñena et al [9] investigated some properties of the Kepler problem related to the existence of quasi-bi-Hamiltonian structures.…”