Physics: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 [math]\displaystyle{ Q\to \mathbb R }[/math], a free motion equation is defined as a second order non-autonomous dynamic equation on [math]\displaystyle{ Q\to \mathbb R }[/math] which is brought into the form
- [math]\displaystyle{ \overline q^i_{tt}=0 }[/math]
with respect to some reference frame [math]\displaystyle{ (t,\overline q^i) }[/math] on [math]\displaystyle{ Q\to \mathbb R }[/math]. Given an arbitrary reference frame [math]\displaystyle{ (t,q^i) }[/math] on [math]\displaystyle{ Q\to \mathbb R }[/math], a free motion equation reads
- [math]\displaystyle{ 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]\displaystyle{ \Gamma^i=\partial_t q^i(t,\overline q^j) }[/math] is a connection on [math]\displaystyle{ Q\to \mathbb R }[/math] associates with the initial reference frame [math]\displaystyle{ (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]\displaystyle{ Q\to\mathbb R }[/math] of a mechanical system is a toroidal cylinder [math]\displaystyle{ T^m\times \mathbb R^k }[/math].
See also
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).
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