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I was wondering, how would one formulate Newtonian Mechanics as a rigorous mathematical model? Would one take force to be an external quantity defined by various force laws (Coulomb, UG) and accept Newton's Second Law of Motion as an axiom (and first law as a definition of inertial frames)? Or would one take the first and second law to be definitons of force (in whitch case it seems to me the concept of inertial frame becomes impossible to define)? Also, would one take mass to be externally defined?

It seems to me both of the approaches, which are mathematically incompatiable, are used in textbooks. What's worse, they are intermixed according to convenience, confusing me about inertial reference frames.

While it seems that later formulations of classical mechanics (Lagrangian, Hamiltonian, Hamilton-Jacobi) are introduced in a rigorous manner, the structure of Newtonian Mechanics is never presented in any detail. I am a fanatic for rigour, proofs and highly abstract and formal statements (even though I prefer physics over mathematics any day). For me this state of ignorance is rather disturbing. Can you help me out?

Thanks.

Molu