Non-autonomous mechanics describe non-relativistic mechanical systems subject to time-dependent transformations. In particular, this is the case of mechanical systems whose Lagrangians and Hamiltonians depend on the time. The configuration space of non-autonomous mechanics is a fiber bundle
Q
→
R
over the time axis
R
coordinated by
(
t
,
q
i
)
.
This bundle is trivial, but its different trivializations
Q
=
R
×
M
correspond to the choice of different non-relativistic reference frames. Such a reference frame also is represented by a connection
Γ
on
Q
→
R
which takes a form
Γ
i
=
0
with respect to this trivialization. The corresponding covariant differential
(
q
t
i
−
Γ
i
)
∂
i
determines the relative velocity with respect to a reference frame
Γ
.
As a consequence, non-autonomous mechanics (in particular, non-autonomous Hamiltonian mechanics) can be formulated as a covariant classical field theory (in particular covariant Hamiltonian field theory) on
X
=
R
. Accordingly, the velocity phase space of non-autonomous mechanics is the jet manifold
J
1
Q
of
Q
→
R
provided with the coordinates
(
t
,
q
i
,
q
t
i
)
. Its momentum phase space is the vertical cotangent bundle
V
Q
of
Q
→
R
coordinated by
(
t
,
q
i
,
p
i
)
and endowed with the canonical Poisson structure. The dynamics of Hamiltonian non-autonomous mechanics is defined by a Hamiltonian form
p
i
d
q
i
−
H
(
t
,
q
i
,
p
i
)
d
t
.
One can associate to any Hamiltonian non-autonomous system an equivalent Hamiltonian autonomous system on the cotangent bundle
T
Q
of
Q
coordinated by
(
t
,
q
i
,
p
,
p
i
)
and provided with the canonical symplectic form; its Hamiltonian is
p
−
H
.
