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 over. 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 of. A dynamic equation on 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. Given bundle coordinates on and the adapted coordinates 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 is defined as a holonomic
connection on a jet bundle. 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 of. A free motion equation in non-autonomous mechanics exemplifies a second-order non-autonomous dynamic equation.
