An alternative derivation of Sraffa’s fundamental equation with applications

I derive Sraffa’s fundamental equation r = R(1 − w) by means of differential equations and optimisation, on which I work three remarks: (i) I analytically provide an alternative formulation of Sraffa’s fundamental equation; (ii) it is analogous to the optimisation problem of a particle moving along a straight line; (iii) the optimisation problem’s objective function is that of the minimisation of R. I additionally ask whether such an optimisation problem may also apply to any corresponding ‘Real System’ of the ‘Standard System’, to which it is already found to apply, and I answer positively. I ulteriorly assess the application of Heisenberg’s Uncertainty Principle to the same equation and derive an equation for the momentum of the particle in terms of its momentum uncertainty and in terms of its position, in which the particle is the ‘Standard Net Product’ and its momentum is R. I finally appraise the brachistochrone problem from a Sraffian perspective and find that in the presence of distributional gravity, for a meaningful mass function for the ‘Standard Net Product’, the optimal path for the distribution of the ‘Standard Net Product’ between profits and wages is no longer r = R(1 − w), but a Sraffian cycloid with specific position coordinates w and r.