The key problem of classical mechanics is that there is no intrinsic definition of an inertial reference frame. We have different inertial reference frames which are not inertial with respect to each other.

Classical Lagrangian and Hamiltonian 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. Given such an equation, different inertial reference frames for this differ from each other in constant velocities.*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)

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