non-autonomous system (mathematics)

{{Multiple issues|

{{context|date=May 2025}}

{{technical|date=May 2025}}

}}In mathematics, an autonomous system is a dynamic equation on a smooth manifold. A non-autonomous system is a dynamic equation on a smooth fiber bundle $Q\backslash to\; \backslash mathbb\; R$ over $\backslash mathbb\; R$. For instance, this is the case of non-autonomous mechanics.

An r-order differential equation on a fiber bundle $Q\backslash to\; \backslash mathbb\; R$ is represented by a closed subbundle of a jet bundle $J^rQ$ of $Q\backslash to\; \backslash mathbb\; R$. A dynamic equation on $Q\backslash to\; \backslash mathbb\; R$ is a differential equation which is algebraically solved for a higher-order derivatives.

In particular, a first-order dynamic equation on a fiber bundle $Q\backslash to\; \backslash mathbb\; R$ is a kernel of the covariant differential of some connection $\backslash Gamma$ on $Q\backslash to\; \backslash mathbb\; R$. Given bundle coordinates $(t,q^i)$ on $Q$ and the adapted coordinates $(t,q^i,q^i\_t)$ on a first-order jet manifold $J^1Q$, a first-order dynamic equation reads

: $q^i\_t=\backslash Gamma\; (t,q^i).$

For instance, this is the case of Hamiltonian non-autonomous mechanics.

A second-order dynamic equation

: $q^i\_\{tt\}=\backslash xi^i(t,q^j,q^j\_t)$

on $Q\backslash to\backslash mathbb\; R$ is defined as a holonomic

connection $\backslash xi$ on a jet bundle $J^1Q\backslash to\backslash mathbb\; R$. This

equation also is represented by a connection on an affine jet bundle $J^1Q\backslash to\; Q$. Due to the canonical

embedding $J^1Q\backslash to\; TQ$, it is equivalent to a geodesic equation

on the tangent bundle $TQ$ of $Q$. A free motion equation in non-autonomous mechanics exemplifies a second-order non-autonomous dynamic equation.