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 → R over R . For instance, this is the case of non-autonomous mechanics.
An r-order differential equation on a fiber bundle Q → R is represented by a closed subbundle of a jet bundle J r Q of Q → R . A dynamic equation on Q → 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 → R is a kernel of the covariant differential of some connection Γ on Q → R . Given bundle coordinates ( t , q i ) on Q and the adapted coordinates ( t , q i , q t i ) on a first-order jet manifold J 1 Q , a first-order dynamic equation reads
q t i = Γ ( t , q i ) . For instance, this is the case of Hamiltonian non-autonomous mechanics.
A second-order dynamic equation
q t t i = ξ i ( t , q j , q t j ) on Q → R is defined as a holonomic connection ξ on a jet bundle J 1 Q → R . This equation also is represented by a connection on an affine jet bundle J 1 Q → Q . Due to the canonical imbedding J 1 Q → T Q , it is equivalent to a geodesic equation on the tangent bundle T Q of Q . A free motion equation in non-autonomous mechanics exemplifies a second-order non-autonomous dynamic equation.
