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 $$Q\to \mathbb R$$, a free motion equation is defined as a second order non-autonomous dynamic equation on $$Q\to \mathbb R$$ which is brought into the form


 * $$\overline q^i_{tt}=0$$

with respect to some reference frame $$(t,\overline q^i)$$ on $$Q\to \mathbb R$$. Given an arbitrary reference frame $$(t,q^i)$$ on $$Q\to \mathbb R$$, a free motion equation reads


 * $$q^i_{tt}=d_t\Gamma^i +\partial_j\Gamma^i(q^j_t-\Gamma^j) -

\frac{\partial q^i}{\partial\overline q^m}\frac{\partial\overline q^m}{\partial q^j\partial q^k}(q^j_t-\Gamma^j) (q^k_t-\Gamma^k),$$

where $$\Gamma^i=\partial_t q^i(t,\overline q^j)$$ is a connection on $$Q\to \mathbb R$$ associates with the initial reference frame $$(t,\overline q^i)$$. 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 $$Q\to\mathbb R$$ of a mechanical system is a toroidal cylinder $$T^m\times \mathbb R^k$$.