In mathematics, a non-autonomous system of ordinary differential equations is defined to be a dynamic equation on a smooth fiber bundle
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
Such a system admits transformations of a coordinate
depending on other coordinates on
Therefore, it is called the relativistic system.
In particular, Special Relativity on the Minkowski space
of a relativistic system has no preferable fibration over
, a velocity space of relativistic system is a first order jet manifold
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
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.
, a first order jet manifold
is provided with the adapted coordinates
possessing transition functions The relativistic velocities of a relativistic system are represented by elements of a fibre bundle
is the tangent bundle of
Then a generic equation of motion of a relativistic system in terms of relativistic velocities reads For instance, if
μ ν
, this is an equation of a relativistic charge in the presence of an electromagnetic field.