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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 , a free motion equation is defined as a second order non-autonomous dynamic equation on which is brought into the form

with respect to some reference frame on . Given an arbitrary reference frame on , a free motion equation reads

where is a connection on associates with the initial reference frame . 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 of a mechanical system is a toroidal cylinder .

- 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).