Newtonian Mechanics from "axioms"

Classical approach

We begin with the following principles-postulates

1. (First Newton law) Among all the possible reference frames there is a class of them, called inertial frames, in which physical laws are in their simplest form.
2. (Second Newton law) In particular, in any inertial frame we have for any particle over which is acting a force $F$:

where $m$ is a constant of the particle that we will call mass.

3. (Third Newton law) For every force there exist other with opposite direction acting in the particle that originated the first one.
4. (Galilean transformation) A particle with constant velocity in an inertial frame originate other one of which is its origin, and reciprocally. To change coordinates between inertial frames we use the Galilean matrices $G_v$, with $v$ being the velocity of one to the other. The physical laws remain the same in different inertial frames.
5. All the forces in Nature come from the gradient of a potencial or, in mathematical terms, every force is an exact differential 1-form in ${\mathbb{R}}^3$. This also fits with the fact that work is an integral along a curve, so the object to be integrated must be a 1-form (see integration on Rn).

From here we can conclude several results, some of them considered, wrongly, "principles", by the physicists:

• The principle of conservation of momentum. Can be derived from the third Newton law. Also, if we started with principle of least action instead, it is concluded from it, expressing the system in the following generalized coordinates: the centre of mass and the distances of every particle to it.
• Conservation of energy. It can be derived from Newton second law and postulate 5 (forces are conservative). It is shown in work and energy. Also, I think is a bit like cheating, because kinetic energy is just defined to get this working.
• Principle of least action. From Newton's laws you can deduce the Euler-Lagrange equations and, therefore, the least action principle. You must assume, too, that all the forces are conservative. Conversely, from principle of least action you could proof Newton laws, so they are equivalent under some assumptions. In the end, it will be taken as the main principle to derive almost everything: in special relativity and general relativity, in Quantum Mechanics,... For that reason, the only interesting part of this equivalence is that least action principle implies Newton law, because it shows that Newtonian mechanics is a particular case of physics. Moreover, Noether's theorem let us see the conservation theorems follow directly from here, from the simmetries of the Lagrangian.
• Conservation of angular momentum. It is concluded using the least action principle applied to the lagrangian of a free particle in polar coordinates.

Schuller approach

Newton's First Two Laws
(I) A body on which no force acts moves uniformly along a straight line.
(II) Deviation of a body’s motion from such straight motion is effected by a force, reduced by a factor of the body’s reciprocal mass.

The first thing we note is that, if read as a prescription of what a body does, the first axiom is merely a specific case of the second one (i.e., just let the force vanish in Newton II). We, therefore, need to read the first axiom in a different manner: you assume that a particle is not experiencing any forces, and you use these particles to experimentally check what a straight line is. The first axiom is a measurement prescription for geometry.

The second important point we need to note is that, if we view gravity as a force, the first axiom is only useful if we consider a universe in which a single particle lives. That is, gravity universally acts on all massive objects, and so if we have two massive particles in our universe (which our Universe clearly does), they must both experience a force, and so Newton (I) becomes useless... unless we stop thinking of gravity as a force.
You might think that we’re being a bit pedantic here and just say ‘oh ok, but we can just use Newton II and go on our merry way!’ The problem with that is that Newton II talks about the deviation from a straight line, and without Newton I we don’t know what a straight line is.

This is the entry to Newton-Cartan gravitation.