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 → R {\displaystyle Q\to \mathbb {R} } , a free motion equation is defined as a second order non-autonomous dynamic equation on Q → R {\displaystyle Q\to \mathbb {R} } which is brought into the form
q ¯ t t i = 0 {\displaystyle {\overline {q}}_{tt}^{i}=0}
with respect to some reference frame ( t , q ¯ i ) {\displaystyle (t,{\overline {q}}^{i})} on Q → R {\displaystyle Q\to \mathbb {R} } . Given an arbitrary reference frame ( t , q i ) {\displaystyle (t,q^{i})} on Q → R {\displaystyle Q\to \mathbb {R} } , a free motion equation reads
q t t i = d t Γ i + ∂ j Γ i ( q t j − Γ j ) − ∂ q i ∂ q ¯ m ∂ q ¯ m ∂ q j ∂ q k ( q t j − Γ j ) ( q t k − Γ k ) , {\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 Γ i = ∂ t q i ( t , q ¯ j ) {\displaystyle \Gamma ^{i}=\partial _{t}q^{i}(t,{\overline {q}}^{j})} is a connection on Q → R {\displaystyle Q\to \mathbb {R} } associates with the initial reference frame ( t , q ¯ i ) {\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
Q → R {\displaystyle Q\to \mathbb {R} } of a mechanical system is a toroidal cylinder T m × R k {\displaystyle 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).