Lie–Hamilton systems on curved spaces: A geometrical approach

Producción CientíficaA Lie–Hamilton system is a nonautonomous system of first-order ordinary differential equations describing the integral curves of a t-dependent vector field taking values in a finite-dimensional Lie algebra, a Vessiot–Guldberg Lie algebra, of Hamiltonian vector fields relative to a Poisson structure. Its general solution can be written as an autonomous function, the superposition rule, of a generic finite family of particular solutions and a set of constants. We pioneer the study of Lie–Hamilton systems on Riemannian spaces (sphere, Euclidean and hyperbolic plane), pseudo-Riemannian spaces (anti-de Sitter, de Sitter, and Minkowski spacetimes) as well as on semi-Riemannian spaces (Newtonian spacetimes). Their corresponding constants of motion and superposition rules are obtained explicitly in a geometric way. This work extends the (graded) contraction of Lie algebras to a contraction procedure for Lie algebras of vector fields, Hamiltonian functions, and related symplectic structures, invariants, and superposition rules

Associated noncommutative vector bundles over the Vaksman–Soibelman quantum complex projective spaces

Analysis and Stochastic

Structure-Sensitive Mechanism of Nanographene Failure

The response of a nanographene sheet to external stresses is considered in terms of a mechanochemical reaction. The quantum chemical realization of the approach is based on a coordinate-of-reaction concept for the purpose of introducing a mechanochemical internal coordinate (MIC) that specifies a deformational mode. The related force of response is calculated as the energy gradient along the MIC, while the atomic configuration is optimized over all of the other coordinates under the MIC constant-pitch elongation. The approach is applied to the benzene molecule and (5, 5) nanographene. A drastic anisotropy in the microscopic behavior of both objects under elongation along a MIC has been observed when the MIC is oriented either along or normally to the C-C bonds chain. Both the anisotropy and high stiffness of the nanographene originate at the response of the benzenoid unit to stress.Comment: 19 pages, 7 figures 1 tabl