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

${\displaystyle {\overline {q}}_{tt}^{i}=0}$

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

${\displaystyle q_{tt}^{i}=d_{t}\Gamma ^{i}+\partial _{j}\Gamma ^{i}(q_{t}^{j}-\Gamma ^{j})-{\frac {\partial q^{i}}{\partial {\overline {q}}^{m}}}{\frac {\partial {\overline {q}}^{m}}{\partial q^{j}\partial q^{k}}}(q_{t}^{j}-\Gamma ^{j})(q_{t}^{k}-\Gamma ^{k}),}$

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