Book suggestions for foundation of Newtonian Mechanics

You may have a look at the works of Noll and Truesdell, for example

Lectures on the foundations of continuum mechanics and thermodynamics

or at the paper of Eisenbud (which maybe has influenced implicitly many textbooks):

On the Classical Laws of Motion .

Note that this articles are written from completely different epistemological points of view.

I didn't read it yet, but this may be of interest, too: Classical Dynamics: A Contemporary Approach.

---

Some other references:

Mechanical systems, classical models By P. P. Teodorescu

Differential Equations, Mechanics, and Computation by R. Palais

Rational Mechanics by C.W.Kilmister

A first course in rational continuum mechanics by C. Truesdell

The elements of continuum mechanics by C. Truesdell

Mathematical aspects of classical and celestial mechanics By V. Arnolʹd, V. Kozlov, A. Neĭshtadt