In mathematics, a non-autonomous system of ordinary differential equations is defined to be a dynamic equation on a smooth fiber bundle
Q
→
R
over
R
. For instance, this is the case of non-relativistic non-autonomous mechanics, but not relativistic mechanics. To describe relativistic mechanics, one should consider a system of ordinary differential equations on a smooth manifold
Q
whose fibration over
R
is not fixed. Such a system admits transformations of a coordinate
t
on
R
depending on other coordinates on
Q
. Therefore, it is called the relativistic system. In particular, Special Relativity on the Minkowski space
Q
=
R
4
is of this type.
Since a configuration space
Q
of a relativistic system has no preferable fibration over
R
, a velocity space of relativistic system is a first order jet manifold
J
1
1
Q
of one-dimensional submanifolds of
Q
. The notion of jets of submanifolds generalizes that of jets of sections of fiber bundles which are utilized in covariant classical field theory and non-autonomous mechanics. A first order jet bundle
J
1
1
Q
→
Q
is projective and, following the terminology of Special Relativity, one can think of its fibers as being spaces of the absolute velocities of a relativistic system. Given coordinates
(
q
0
,
q
i
)
on
Q
, a first order jet manifold
J
1
1
Q
is provided with the adapted coordinates
(
q
0
,
q
i
,
q
0
i
)
possessing transition functions
q
′
0
=
q
′
0
(
q
0
,
q
k
)
,
q
′
i
=
q
′
i
(
q
0
,
q
k
)
,
q
′
0
i
=
(
∂
q
′
i
∂
q
j
q
0
j
+
∂
q
′
i
∂
q
0
)
(
∂
q
′
0
∂
q
j
q
0
j
+
∂
q
′
0
∂
q
0
)
−
1
.
The relativistic velocities of a relativistic system are represented by elements of a fibre bundle
R
×
T
Q
, coordinated by
(
τ
,
q
λ
,
a
τ
λ
)
, where
T
Q
is the tangent bundle of
Q
. Then a generic equation of motion of a relativistic system in terms of relativistic velocities reads
(
∂
λ
G
μ
α
2
…
α
2
N
2
N
−
∂
μ
G
λ
α
2
…
α
2
N
)
q
τ
μ
q
τ
α
2
⋯
q
τ
α
2
N
−
(
2
N
−
1
)
G
λ
μ
α
3
…
α
2
N
q
τ
τ
μ
q
τ
α
3
⋯
q
τ
α
2
N
+
F
λ
μ
q
τ
μ
=
0
,
G
α
1
…
α
2
N
q
τ
α
1
⋯
q
τ
α
2
N
=
1.
For instance, if
Q
is the Minkowski space with a Minkowski metric
G
μ
ν
, this is an equation of a relativistic charge in the presence of an electromagnetic field.