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->*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***R**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.

**References:**

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**

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