Dualities in Nonholonomic Optimization

This article deals with optimizing problems whose restrictions are nonholonomic. The central issue relates to dual nonholonomic programs (what they mean and how they are solved?) when the nonholonomic constraints are given by Pfaff equations. We emphasize that nonholonomic critical points are not the classical ones and that the nonholonomic Lagrange multipliers are not the classical (holonomic) Lagrange multipliers. Topological significance of Lagrange multipliers and dual function theory introduced by EDO and EDP are key results. Also new Riemannian geometries attached to a given nonholonomic constrained optimization problem are introduced. The original results are surprising and include: (i) aspects derived from the Vranceanu theory of nonholonomic manifolds, and from the geometric distributions theory, (ii) optimal problems in Darboux canonical coordinates.

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

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.