A System of n = 3 Coupled Oscillators with Magnetic Terms: Symmetries and Integrals of Motion

Abstract

Abstract. The properties of a system of n = 3 coupled oscillators with linear terms in the velocities (magnetic terms) depending in two parameters are studied. We proved the existence of a bi-Hamiltonian structure arising from a non-symplectic symmetry, as well the existence of master symmetries and additional integrals of motion (weak superintegrability) for certain particular values of the parameters.

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

“…The purpose is to study new and interesting properties of the Kepler problem. In fact, it has been proved that if a dynamical vector field satisfies certain properties (existence of canonoid transformations [9,10] or existence of non-symplectic symmetries [27,28]) then it is Hamiltonian with respect to two different structures without satisfying necessarily the Nijenhuis condition.…”

Hamiltonian Dynamics of a spaceship in Alcubierre and Gödel metrics: Recursion operators and underlying master symmetries

“…is called a master symmetry or a generator of symmetries of degree m = 1 for the Hamiltonian vector field X H [7,8,10,27,28].…”

“…This is well illustrated in Fig 1 . In differential geometric terms, Y j and Hj are called master symmetries for X i and master integrals, respectively, [7,8,10,27,28].…”

“…In 1997 and 1999, Smirnov [30,31] formulated a constructive method of transforming a completely integrable Hamiltonian system, in Liou-ville's sense, into Magri-Morosi-Gel'fand-Dorfman's (MMGD) bi-Hamiltonian form. In 2005, Rañada [27] 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 cases. Recently, in 2021 [19], we also constructed a hierarchy of bi-Hamiltonian structures for the Kepler problem, and computed conserved quantities using related master symmetries.…”

Noncommutative Kepler Dynamics: symmetry groups and bi-Hamiltonian structures

“…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].…”