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.
