Relativistic system (mathematics)

In mathematics, a non-autonomous system of ordinary differential equations is defined to be a dynamic equation on a smooth fiber bundle formula_1 over formula_2. 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 formula_3 whose fibration over formula_2 is not fixed. Such a system admits transformations of a coordinate formula_5 on formula_2 depending on other coordinates on formula_3. Therefore, it is called the relativistic system. In particular, Special Relativity on the
Minkowski space formula_8 is of this type.
Since a configuration space formula_3 of a relativistic system has no
preferable fibration over formula_2, a 
velocity space of relativistic system is a first order jet
manifold formula_11 of one-dimensional submanifolds of formula_3. 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 formula_13 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 formula_14 on formula_3, a first order jet manifold formula_11 is provided with the adapted coordinates formula_17 
possessing transition functions 
The relativistic velocities of a relativistic system are represented by
elements of a fibre bundle formula_19, coordinated by formula_20, where formula_21 is the tangent bundle of formula_3. Then a generic equation of motion of a relativistic system in terms of relativistic velocities reads
For instance, if formula_3 is the Minkowski space with a Minkowski metric formula_26, this is an equation of a relativistic charge in the presence of an electromagnetic field.