Non-autonomous system (mathematics)

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 $$Q\to \mathbb R$$ over $$\mathbb R$$. For instance, this is the case of non-autonomous mechanics.

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

In particular, a first-order dynamic equation on a fiber bundle $$Q\to \mathbb R$$ is a kernel of the covariant differential of some connection $$\Gamma$$ on $$Q\to \mathbb R$$. 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


 * $$q^i_t=\Gamma (t,q^i).$$

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

A second-order dynamic equation


 * $$q^i_{tt}=\xi^i(t,q^j,q^j_t)$$

on $$Q\to\mathbb R$$ is defined as a holonomic connection $$\xi$$ on a jet bundle $$J^1Q\to\mathbb R$$. This equation also is represented by a connection on an affine jet bundle $$J^1Q\to Q$$. Due to the canonical embedding $$J^1Q\to TQ$$, 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.