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}}
.

• Non-autonomous mechanics
• Non-autonomous system (mathematics)
• Analytical mechanics
• Fictitious force

• De Leon, M., Rodrigues, P., Methods of Differential Geometry in Analytical Mechanics (North Holland, 1989).
• Giachetta, G., Mangiarotti, L., Sardanashvily, G., Geometric Formulation of Classical and Quantum Mechanics (World Scientific, 2010) ISBN 981-4313-72-6 (arXiv:0911.0411).