On the last post, I have established idea of the reparametrization of force by s. To rigorously define such function, we would have to, of course, resort to pure mathematical machinery. I am working on this, with all the existence conditions and further property, and I will share a few details.
We are familiar with the Stieljtes Integral, which generalizes the notions of Riemann and Lebesgue Integrals. If we have an integral of the form
P = int[f(x)] dx
and given that x = h(u), Then we write the integral above as
P = int[ f(h(u)) ] dh(u)
as long as the related sum exists. Base on this, we shall define a similar integral. Consider an integral
P = int [ f(v) ] dv.
Let P = dW/dt (which implies W = int [P] dt). Thus we have now
W = int int [ f(v) ] dv dt.
Consider there exist a bijective function u = u(t). Also define u'(t) = v(t). Because of the bijectivity, we can have f(v(u)) = h(u). Thus
W = int int [ h(u) ] dv dt.
Now I have two ways evaluate W. But I will not discuss it here ( I will try to get the paper on this done and put it on arxiv.org). The integral above reduces to
W = int [h(u)] du.
For now, I know that for the last equation to exist, u = u(t) must be bijective and must be at least of the class C. The details of this derivations is at this current moment rather incomplete, in the sense that it is not totally rigorous. I am working at my best to finish this, perhaps by the end of this year end holiday.