Английская Википедия:Free motion equation

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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 <math>Q\to \mathbb R</math>, a free motion equation is defined as a second order non-autonomous dynamic equation on <math>Q\to \mathbb R</math> which is brought into the form

<math>\overline q^i_{tt}=0</math>

with respect to some reference frame <math>(t,\overline q^i)</math> on <math>Q\to \mathbb R</math>. Given an arbitrary reference frame <math>(t,q^i)</math> on <math>Q\to \mathbb R</math>, a free motion equation reads

<math>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),</math>

where <math>\Gamma^i=\partial_t q^i(t,\overline q^j)</math> is a connection on <math>Q\to \mathbb R</math> associates with the initial reference frame <math>(t,\overline q^i)</math>. 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 <math>Q\to\mathbb R</math> of a mechanical system is a toroidal cylinder <math>T^m\times \mathbb R^k</math>.

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) Шаблон:ISBN (Шаблон:ArXiv).

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