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 , a free motion equation is defined as a second order non-autonomous dynamic equation on Q → R which is brought into the form
q ¯ t t i = 0 with respect to some reference frame ( t , q ¯ i ) on Q → R . Given an arbitrary reference frame ( t , q i ) on Q → R , a free motion equation reads
q t t i = d t Γ i + ∂ j Γ i ( q t j − Γ j ) − ∂ q i ∂ q ¯ m ∂ q ¯ m ∂ q j ∂ q k ( q t j − Γ j ) ( q t k − Γ k ) , where Γ i = ∂ t q i ( t , q ¯ j ) is a connection on Q → R associates with the initial reference frame ( t , q ¯ i ) . 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 Q → R of a mechanical system is a toroidal cylinder T m × R k .
