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

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

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

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

where $\Gamma ^{i}=\partial _{t}q^{i}(t,{\overline {q}}^{j})$ 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 $T^{m}\times \mathbb {R} ^{k}$.

See also

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

References

• 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).