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
, a free motion equation is defined as a second order non-autonomous dynamic equation on
which is brought into the form with respect to some reference frame
Given an arbitrary reference frame
, a free motion equation reads where
associates with the initial reference frame
The right-hand side of this equation is treated as an inertial force.
A free motion equation need not exist in general.
of a mechanical system is a toroidal cylinder
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