Free motion equation

A free motion equation is a differential equation that describes a mechanical system in the absence of external forces, but in the presence only of an inertial force depending on the choice of a reference frame. 
In non-autonomous mechanics on a configuration space formula_1, a free motion equation is defined as a second order non-autonomous dynamic equation on formula_1 which is brought into the form 
with respect to some reference frame formula_4 on formula_1. Given an arbitrary reference frame formula_6 on formula_1, a free motion equation reads
where formula_9 is a connection on formula_1 associates with the initial reference frame formula_4. The right-hand side of this equation is treated as an inertial force. 
A free motion equation need not exist in general. It can be defined if and only if a configuration bundle 
formula_12 of a mechanical system is a toroidal cylinder formula_13.