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
{\displaystyle Q\to \mathbb {R} }
over
R
{\displaystyle \mathbb {R} }
. For instance, this is the case of non-autonomous mechanics.
An r-order differential equation on a fiber bundle
Q
→
R
{\displaystyle Q\to \mathbb {R} }
is represented by a closed subbundle of a jet bundle
J
r
Q
{\displaystyle J^{r}Q}
of
Q
→
R
{\displaystyle Q\to \mathbb {R} }
. A dynamic equation on
Q
→
R
{\displaystyle Q\to \mathbb {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
{\displaystyle Q\to \mathbb {R} }
is a kernel of the covariant differential of some connection
Γ
{\displaystyle \Gamma }
on
Q
→
R
{\displaystyle Q\to \mathbb {R} }
. Given bundle coordinates
(
t
,
q
i
)
{\displaystyle (t,q^{i})}
on
Q
{\displaystyle Q}
and the adapted coordinates
(
t
,
q
i
,
q
t
i
)
{\displaystyle (t,q^{i},q_{t}^{i})}
on a first-order jet manifold
J
1
Q
{\displaystyle J^{1}Q}
, a first-order dynamic equation reads
q
t
i
=
Γ
(
t
,
q
i
)
.
{\displaystyle q_{t}^{i}=\Gamma (t,q^{i}).}
For instance, this is the case of Hamiltonian non-autonomous mechanics.
A second-order dynamic equation
a) Proof:
Consider the equations ## \dot{a}=-\epsilon\sin\theta h(a\cos\theta, a\sin\theta) ## and ## \dot{\theta}=-1-\frac{\epsilon}{a}\cos\theta h(a\cos\theta, a\sin\theta) ##.
Let ## \theta(t)=\psi(t)-t ##.
Then ## \dot{\theta}(t)=\dot{\psi}(t)-1 ##.
By direct substitution of ##...