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

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

with respect to some reference frame $(t,\stackrel{i}{\stackrel{‾}{q}})$ on $Q\to \mathbb{ℝ}$. Given an arbitrary reference frame $(t,q^i)$ on $Q\to \mathbb{ℝ}$, a free motion equation reads

$q_{tt}^i=d_t{\mathrm{\Gamma }}^i+{\partial }_j{\mathrm{\Gamma }}^i(q_t^j-{\mathrm{\Gamma }}^j)-\frac{\partial q^i}{\partial \stackrel{m}{\stackrel{‾}{q}}}\frac{\partial \stackrel{m}{\stackrel{‾}{q}}}{\partial q^j\partial q^k}(q_t^j-{\mathrm{\Gamma }}^j)(q_t^k-{\mathrm{\Gamma }}^k),$

where ${\mathrm{\Gamma }}^i={\partial }_t{q}^i(t,\stackrel{j}{\stackrel{‾}{q}})$ is a connection on $Q\to \mathbb{ℝ}$ associates with the initial reference frame $(t,\stackrel{i}{\stackrel{‾}{q}})$. 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).