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 Q → R {\displaystyle Q\to \mathbb {R} } , a free motion equation is defined as a second order non-autonomous dynamic equation on Q → R {\displaystyle Q\to \mathbb {R} } which is brought into the form q ¯ t t i = 0 {\displaystyle {\overline {q}}_{tt}^{i}=0} with respect to some reference frame ( t , q ¯ i ) {\displaystyle (t,{\overline {q}}^{i})} on Q → R {\displaystyle Q\to \mathbb {R} } .