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 formula_1 over formula_2. For instance, this is the case of non-autonomous mechanics.
An "r"-order differential equation on a fiber bundle formula_1 is represented by a closed subbundle of a jet bundle formula_4 of formula_1. A dynamic equation on formula_1 is a differential equation which is algebraically solved for a higher-order derivatives.
In particular, a first-order dynamic equation on a fiber bundle formula_1 is a kernel of the covariant differential of some connection formula_8 on formula_1. Given bundle coordinates formula_10 on formula_11 and the adapted coordinates formula_12 on a first-order jet manifold formula_13, a first-order dynamic equation reads
For instance, this is the case of Hamiltonian non-autonomous mechanics.
A second-order dynamic equation
on formula_16 is defined as a holonomic
connection formula_17 on a jet bundle formula_18. This
equation also is represented by a connection on an affine jet bundle formula_19. Due to the canonical
embedding formula_20, it is equivalent to a geodesic equation
on the tangent bundle formula_21 of formula_11. A free motion equation in non-autonomous mechanics exemplifies a second-order non-autonomous dynamic equation.