Discrete Mechanics in Nonuniform Time \(\alpha \) -Lattices

In this chapter, we consider special types of nonuniform time lattices by introducing the \(\alpha \) -addition and \(\alpha \) -subtraction from the point of view of pseudo-analysis, a generalization of the classical analysis, where, instead of the field of real numbers, a semiring is taken on a real interval \([a, b] \subset [-\infty , +\infty ].\) In this framework, we deduce related algebraic operations and discuss their properties. We define the discrete conformable fractional derivative and integral by introducing the fractional falling factorial. We arrive at a formulation of discrete mechanics and discuss relevant examples of one-dimensional discrete mechanics in time \(\alpha \) -lattice. We also derive and discuss the Langevin equation.