free motion equation

{{technical|date=May 2025}}

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 $Q\backslash to\; \backslash mathbb\; R$, a free motion equation is defined as a second order non-autonomous dynamic equation on $Q\backslash to\; \backslash mathbb\; R$ which is brought into the form

: $\backslash overline\; q^i\_\{tt\}=0$

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

: $q^i\_\{tt\}=d\_t\backslash Gamma^i\; +\backslash partial\_j\backslash Gamma^i(q^j\_t-\backslash Gamma^j)\; -$

\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 $\backslash Gamma^i=\backslash partial\_t\; q^i(t,\backslash overline\; q^j)$ is a connection on $Q\backslash to\; \backslash mathbb\; R$ associates with the initial reference frame $(t,\backslash overline\; q^i)$. 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

$Q\backslash to\backslash mathbb\; R$ of a mechanical system is a toroidal cylinder $T^m\backslash times\; \backslash mathbb\; R^k$.