Deformed special relativity based on $\alpha$-deformed binary operations

Abstract

In this paper, we define a new velocity having a dimension of $(Length)^{\alpha}/(Time)$ and a new acceleration having a dimension of $(Length)^{\alpha}/(Time)^2$, based on the fractional addition rule. We then discuss the fractional mechanics in one dimension. We show the conservation of fractional energy, and formulate the Hamiltonian formalism for the fractional mechanics. As a matter of illustration, we exhibit some examples for the fractional mechanics.