What actually are Newton's axioms of classical mechanics

[V0ODO0CH1LD]

Just like the Euler-Lagrange differential equation
$$ \frac{\partial{\mathcal{L}}}{\partial{q}} = \frac{d}{dt}\frac{\partial{\mathcal{L}}}{\partial{\dot{q}}} $$
the hamiltonian equations
$$ \frac{\partial{H}} {\partial{q}} = -\dot{p} $$
$$ \frac{\partial{H}} {\partial{p}} = \dot{q} $$
and the poisson equation
$$ \left\{F,H\right\} = \dot{F} $$
are axioms to different formulations of classical mechanics. What are the axioms of the Newtonian formulation?

Are his three laws separate statements that together make up the axiom? Or are the laws the actual axioms? In which case, can the Newtonian formulation of classical mechanics be explicitly formulated in "standard" mathematical language?

EDIT: by non-"standard" I mean like the fact that $F_{12}=-F_{21}$ makes no sense if not followed by subtitles. And that it only makes sense in a very specific context..

 

[Simon Bridge]

I don't think Newton worked axiomatically - in Principia he used Euclids axioms for geometry implicitly, and used that framework to describe/record observations and the results of experiments.