A new fractional mechanics based on fractional addition

Abstract

In this paper, we introduce a new fractional derivative to define a new fractional velocity and a new fractional acceleration with the fractional space translation symmetry, which is given by fractional addition. We also construct the fractional version for Newton mechanics with fractional space translation symmetry in one dimension. We show the conservation of fractional energy and formulate the fractional Hamiltonian formalism for the fractional mechanics with fractional space translation symmetry. We exhibit some examples for the fractional mechanics with fractional space translation symmetry.