Time-Scale Calculus, is a theory which unifies ordinary (plus fractional and q-) calculus with discrete (and finite differences) calculus.
In a sense, in a similar way the Lebesgue integral (or Riemann–Stieltjes integral) unifies both continous integration and discrete summation.
The rationale is to use time-scales (or measure chains) which include continous (dense) points and discrete or semi-discrete (scattered, isolated) points.
Even Variational Calculus (and Euler-Lagrange equations) and Noether's Theorem have been (recently) generalised (more or less) to time-scale calculus.
Any references to (unified) formulation of Mechanics (Classical, Quantum, Statistical) in time-scale calculus (and their relevance)?
A initial survey paper (2002) on "R. Agarwal, M. Bohner, D. O’Regan and A. Peterson, Dynamic equations on time scales: a
survey, J. Comput. Appl. Math. 141 (2002), no. 1-2, 1–26."