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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)?

UPDATE:

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."

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  • $\begingroup$ Premise: I don't know this calculus very well. However I don't see how it may unify the formulation of e.g. classical and quantum mechanics. The dynamics is in both cases a differential equation (not a difference one). There are other mathematical aspects, in my opinion, that emphasize the relevant differences between the two theories (e.g. commutativity vs non-commutativity, non-linear vs linear dynamics). $\endgroup$ – yuggib Nov 7 '14 at 9:23
  • $\begingroup$ @yuggib, one can say the main difference between clasical/quantum is the finite value of the planck constant $h$ (e.g Ehrenfest theorem). Furthermore there are indeed formal mechanics which are the same for both Classical/Quantum (and Statistical) (this is another question). In any case, "unified" is added extra, even mechanics (classical) references and their relevance is good $\endgroup$ – Nikos M. Nov 7 '14 at 9:39

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