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
For instance, this is the case of non-autonomous mechanics.
An r-order differential equation on a fiber bundle
is represented by a closed subbundle of a jet bundle
is a differential equation which is algebraically solved for a higher-order derivatives.
In particular, a first-order dynamic equation on a fiber bundle
is a kernel of the covariant differential of some connection
on a first-order jet manifold
, a first-order dynamic equation reads For instance, this is the case of Hamiltonian non-autonomous mechanics.
A second-order dynamic equation on
This equation also is represented by a connection on an affine jet bundle
Due to the canonical embedding
, it is equivalent to a geodesic equation on the tangent bundle
A free motion equation in non-autonomous mechanics exemplifies a second-order non-autonomous dynamic equation.