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 on . Given an arbitrary reference frame on , a free motion equation reads
where is a connection on 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. It can be defined if and only if a configuration bundle of a mechanical system is a toroidal cylinder .