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.
