Fractional Almost Kahler - Lagrange Geometry

TL;DR

This paper develops a fractional Lagrange geometry framework using Caputo derivatives, linking it to nonholonomic almost Kahler structures, with implications for quantization and physics.

Contribution

It introduces a novel fractional geometric approach to Lagrange and Finsler structures using Caputo derivatives and constructs compatible almost symplectic forms and connections.

Findings

Constructed fractional almost Kahler structures from Lagrange functions

Established geometric objects for fractional Lagrange geometries

Discussed applications in deformation quantization and physics

Abstract

The goal of this paper is to encode equivalently the fractional Lagrange dynamics as a nonholonomic almost Kahler geometry. We use the fractional Caputo derivative generalized for nontrivial nonlinear connections (N-connections) originally introduced in Finsler geometry, with further developments in Lagrange and Hamilton geometry and, in our approach, with fractional derivatives. For fundamental geometric objects induced canonically by regular Lagrange functions, we construct compatible almost symplectic forms and linear connections completely determined by a "prime" Lagrange (in particular, Finsler) generating function. We emphasize the importance of such constructions for deformation quantization of fractional Lagrange geometries and applications in modern physics.