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, 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.
