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 Q → R {\displaystyle Q\to \mathbb {R} } over R {\displaystyle \mathbb {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 {\displaystyle Q} whose fibration over R {\displaystyle \mathbb {R} } is not fixed. Such a system admits transformations of a coordinate t {\displaystyle t} on R {\displaystyle \mathbb {R} } depending on other coordinates on Q {\displaystyle Q}. Therefore, it is called the relativistic system. In particular, Special Relativity on the

Minkowski space Q = R 4 {\displaystyle Q=\mathbb {R} ^{4}} is of this type.

Since a configuration space Q {\displaystyle Q} of a relativistic system has no preferable fibration over R {\displaystyle \mathbb {R} }, a velocity space of relativistic system is a first order jet manifold J 1 1 Q {\displaystyle J_{1}^{1}Q} of one-dimensional submanifolds of Q {\displaystyle 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 {\displaystyle J_{1}^{1}Q\to 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 ) {\displaystyle (q^{0},q^{i})} on Q {\displaystyle Q}, a first order jet manifold J 1 1 Q {\displaystyle J_{1}^{1}Q} is provided with the adapted coordinates ( q 0 , q i , q 0 i ) {\displaystyle (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 . {\displaystyle q'^{0}=q'^{0}(q^{0},q^{k}),\quad q'^{i}=q'^{i}(q^{0},q^{k}),\quad {q'}_{0}^{i}=\left({\frac {\partial q'^{i}}{\partial q^{j}}}q_{0}^{j}+{\frac {\partial q'^{i}}{\partial q^{0}}}\right)\left({\frac {\partial q'^{0}}{\partial q^{j}}}q_{0}^{j}+{\frac {\partial q'^{0}}{\partial q^{0}}}\right)^{-1}.}

The relativistic velocities of a relativistic system are represented by elements of a fibre bundle R × T Q {\displaystyle \mathbb {R} \times TQ}, coordinated by ( τ , q λ , a τ λ ) {\displaystyle (\tau ,q^{\lambda },a_{\tau }^{\lambda })}, where T Q {\displaystyle TQ} is the tangent bundle of Q {\displaystyle 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 , {\displaystyle \left({\frac {\partial _{\lambda }G_{\mu \alpha _{2}\ldots \alpha _{2N}}}{2N}}-\partial _{\mu }G_{\lambda \alpha _{2}\ldots \alpha _{2N}}\right)q_{\tau }^{\mu }q_{\tau }^{\alpha _{2}}\cdots q_{\tau }^{\alpha _{2N}}-(2N-1)G_{\lambda \mu \alpha _{3}\ldots \alpha _{2N}}q_{\tau \tau }^{\mu }q_{\tau }^{\alpha _{3}}\cdots q_{\tau }^{\alpha _{2N}}+F_{\lambda \mu }q_{\tau }^{\mu }=0,}

G α 1 … α 2 N q τ α 1 ⋯ q τ α 2 N = 1. {\displaystyle G_{\alpha _{1}\ldots \alpha _{2N}}q_{\tau }^{\alpha _{1}}\cdots q_{\tau }^{\alpha _{2N}}=1.}

For instance, if Q {\displaystyle Q} is the Minkowski space with a Minkowski metric G μ ν {\displaystyle G_{\mu \nu }}, this is an equation of a relativistic charge in the presence of an electromagnetic field.

Relativistic system (mathematics)

See also