The key problem of classical mechanics is that there is no intrinsic definition of an inertial reference frame.
Classical non-relativistic mechanics admits the adequate mathematical formulation in terns of fibre bundle Q->R over the time axis R. In this framework, a reference frame is defined as a trivialization of this fibre bundle or, equivalently, as a connection on Q->R.
A second order dynamic equation is called a free motion equation if it can be brought into the form of a zero acceleration ddq/dtdt=0 with respect to some reference frame, and this reference frame is said to be inertial for this equation. Thus a definition of an inertial frame depends on the choice of a free motion equation.
A problem is that, given a different free motion equation ddq’/dtdt=0, an inertial reference frame for it fails to be so the first free motion equation ddq/dtdt=0, and their relative velocity is not constant.
In view of this problem, one should write dynamic equations of non-relativistic mechanics in terms of relative velocities and accelerations with respect to an arbitrary reference frame. However, in this case the strict mathematical notions of a relative acceleration and a non-inertial force are rather sophisticated.
G.Sardanashvily, Relative non-relativistic mechanics, arXiv: 0708.2998
G.Giachetta, L.Mangiarotti and G.Sardanashvily, Geometric Formulation of Classical and Quantum Mechanics (WS, 2010)
WikipediA: Free motion equation