Constant curvature coefficients and exact solutions in fractional gravity and geometric mechanics

Abstract

Abstract:We present a study of fractional configurations in gravity theories and Lagrange mechanics. The approach is based on a Caputo fractional derivative which gives zero for actions on constants. We elaborate fractional geometric models of physical interactions and we formulate a method of nonholonomic deformations to other types of fractional derivatives. The main result of this paper consists of a proof that, for corresponding classes of nonholonomic distributions, a large class of physical theories are modelled as nonholonomic manifolds with constant matrix curvature. This allows us to encode the fractional dynamics of interactions and constraints into the geometry of curve flows and solitonic hierarchies.

Fractional Analogous Models in Mechanics and Gravity Theories

“…[1,2,3,5]. Our geometric arena consists from an abstract fractional manifold α V (we shall use also the term "fractional space" as an equivalent one enabled with certain fundamental geometric structures) with prescribed nonholonomic distribution modeling both the fractional calculus and the non-integrable dynamics of interactions.…”

“…, where we put the label L in order to emphasize that such geometric objects are induced by a fractional Lagrangian as we provided in [1,2,3,5]. We also note that it is possible to "arrange" on…”

“…α ′ β ′ } is described by constant matrix coefficients, see details in [9,10], for integer dimensions, and [5], for fractional dimensions.…”

“…In this paper, we outline some key constructions for analogous classical and quantum fractional theories [1,2,3,4,5,6] when methods of nonholonomic and Lagrange-Finsler geometry are generalized to fractional dimensions. 1 An important consequence of such geometric approaches is that using analogous and bi-Hamilton models (see integer dimension constructions [7,9,10]) and related solitonic systems we can study analytically and numerically, as well to try to construct some analogous mechanical and gravitational systems, with the aim to mimic a nonlinear/fractional nonholonomic dynamics/evolution and even to provide certain schemes of quantization, like in the "fractional" Fedosov approach [4,8].…”

Fractional Exact Solutions and Solitons in Gravity

“…Recently, we extended the fractional calculus to Ricci flow theory, gravity and geometric mechanics, solitonic hierarchies etc [1,2,3,4,5,6]. In this work, we outline some basic geometric constructions related to fractional derivatives and integrals and their applications in modern physics and mechanics.…”