Relativistic system (mathematics)

{{technical|date=May 2025}}In mathematics, a non-autonomous system of ordinary differential equations is defined to be a dynamic equation on a smooth fiber bundle $Q\backslash to\; \backslash mathbb\; R$ over $\backslash mathbb\; R$. 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 $Q$ whose fibration over $\backslash mathbb\; R$ is not fixed. Such a system admits transformations of a coordinate $t$ on $\backslash mathbb\; R$ depending on other coordinates on $Q$. Therefore, it is called the relativistic system. In particular, Special Relativity on the

Minkowski space $Q=\; \backslash mathbb\; R^4$ is of this type.

Since a configuration space $Q$ of a relativistic system has no

preferable fibration over $\backslash mathbb\; R$, a

velocity space of relativistic system is a first order jet

manifold $J^1\_1Q$ 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 $J^1\_1Q\backslash to$

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 $(q^0,\; q^i)$ on $Q$, a first order jet manifold $J^1\_1Q$ is provided with the adapted coordinates $(q^0,q^i,q^i\_0)$

possessing transition functions

: $q\text{'}^0=q\text{'}^0(q^0,q^k),\; \backslash quad\; q\text{'}^i=q\text{'}^i(q^0,q^k),\; \backslash quad$

{q'}^i_0 = \left(\frac{\partial q'^i}{\partial q^j} q^j_0 + \frac{\partial q'^i}{\partial

q^0} \right) \left(\frac{\partial q'^0}{\partial q^j} q^j_0 + \frac{\partial q'^0}{\partial q^0}

\right)^{-1}.

The relativistic velocities of a relativistic system are represented by

elements of a fibre bundle $\backslash mathbb\; R\backslash times\; TQ$, coordinated by $(\backslash tau,q^\backslash lambda,a^\backslash lambda\_\backslash tau)$, where $TQ$ is the tangent bundle of $Q$. Then a generic equation of motion of a relativistic system in terms of relativistic velocities reads

: $\backslash left(\backslash frac\{\backslash partial\_\backslash lambda\; G\_\{\backslash mu\backslash alpha\_2\backslash ldots\backslash alpha\_\{2N\}\}\}\{2N\}-\; \backslash partial\_\backslash mu$

G_{\lambda\alpha_2\ldots\alpha_{2N}}\right) q^\mu_\tau q^{\alpha_2}_\tau\cdots

q^{\alpha_{2N}}_\tau - (2N-1)G_{\lambda\mu\alpha_3\ldots\alpha_{2N}}q^\mu_{\tau\tau} q^{\alpha_3}_\tau\cdots

q^{\alpha_{2N}}_\tau + F_{\lambda\mu}q^\mu_\tau =0,

: $G\_\{\backslash alpha\_1\backslash ldots\backslash alpha\_\{2N\}\}q^\{\backslash alpha\_1\}\_\backslash tau\backslash cdots\; q^\{\backslash alpha\_\{2N\}\}\_\backslash tau=1.$

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