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

$\overline q^i_{tt}=0$

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

$q^i_{tt}=d_t\Gamma^i +\partial_j\Gamma^i(q^j_t-\Gamma^j) - \frac{\partial q^i}{\partial\overline q^m}\frac{\partial\overline q^m}{\partial q^j\partial q^k}(q^j_t-\Gamma^j) (q^k_t-\Gamma^k),$

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 $T^m\times \mathbb R^k$.

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).