Poisson-Lie groups, bi-Hamiltonian systems and integrable deformations

Given a Lie-Poisson completely integrable bi-Hamiltonian system, in the new paper arXiv:1609.07438 we present a method which allows us to construct, under certain conditions, a completely integrable bi-Hamiltonian deformation of the initial Lie-Poisson system on a non-abelian Poisson-Lie group. Moreover, we show that from the two multiplicative (Poisson-Lie) Hamiltonian structures that underly the dynamics of the deformed system and by making use of the non-abelian group law, one may obtain two completely integrable Hamiltonian systems on the direct product of the non-abelian group by itself. By construction, both systems admit reduction, via the multiplication in the non-abelian group, to the initial deformed bi-Hamiltonian system. The previous approach is applied to two relevant Lie-Poisson completely integrable bi-Hamiltonian systems: the Lorenz and Euler top systems.

On Hamiltonians with position-dependent mass from Kaluza-Klein compactifications

In a recent paper (J.R. Morris, Quant. Stud. Math. Found. 2 (2015) 359), an inhomogeneous compactification of the extra dimension of a five-dimensional Kaluza-Klein metric has been shown to generate a position-dependent mass (PDM) in the corresponding four-dimensional system. As an application of this dimensional reduction mechanism, a specific static dilatonic scalar field has been connected with a PDM Lagrangian describing a well-known nonlinear PDM oscillator. In this paper (arXiv:1605.06829) we present more instances of this construction that lead to PDM systems with radial symmetry, and the properties of their corresponding inhomogeneous extra dimensions are compared with the ones in the nonlinear oscillator model. Finally, it is shown that the compactification introduced in this type of models can alternatively be interpreted as a novel mechanism for the dynamical generation of curvature.