As Gugg recommended, in his comment, Archimedes certainly applied mathematics to physics. Whether you're interested in his On Floating Bodies or Method, which contains the first explicit use of what today we'd call infinitesimals, there is no doubt that he applied mathematics to physics. As far as cosmological physicists (which your original question excluded), there are many. Of course, Ptolemy is said to have "discovered" what we would consider trigonometry (though he didn't use trig functions, as symbolic mathematics had not yet been implemented).
But I'd suggest looking into the introduction of symbolic mathematics (algebra). However, symbolic mathematics came about during Galileo's time (but obviously symbolic algebra was not applied to Galileo's work until after his death). For a sustained discussion of the beginnings of symbolic mathematics, I'd suggest reading Jacob Klein's Greek Mathematics and the Origin of Algebra, which includes, in its appendix, a translation of Francois Viete's New Algebra (a work that Descartes "borrowed" greatly from). (Klein's book, however, is not an easy read.) Anyway, although this book does not focus on the exact topic you're asking about, it is pertinent to the question of how mathematical physics (from Descartes to the present) came to fruition; and, in my opinion, it is one of the hidden philosophical gems of the 20th Century.
I should add that Klein's book primarily addresses what he considers to be a great gulf between Greek and modern mathematics. Whether you buy the contention that Greek mathematics was fundamentally different from modern mathematics is up to you; but Klein makes thoughtful and interesting points. (His thoughts are clearly descendants of Husserl's.) But, if I understand your question properly, Klein's book will be quite helpful in trying to understand "what happened" or "what was different" between the ancients and the moderns; the contention that mathematics was fundamentally different is a very interesting answer to those questions. Of course, the contention assumes that ancients like Archimedes were, in fact, applying mathematics to physics; and thus, the difference between an Archimedes and a Descartes (assuming that Galileo would be a sort of interim character) would be symbolic mathematics and/or time/hindsight.
But the use of symbolic mathematics may have been coincident to some fundamental shift in how science was done--or may have been quite important (as it is with modern physics). Of course, a significant change may not have occurred. Or, perhaps, an external factor (to the practice of science) may have been primarily responsible for post-Renaissance science.
There is no reason for an "essential difference" between pre-Renaissance science and post.