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 $Q\to \mathbb {R}$ over $\mathbb {R}$ . 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 $\mathbb {R}$ is not fixed. Such a system admits transformations of a coordinate $t$ on $\mathbb {R}$ depending on other coordinates on $Q$ . Therefore, it is called the relativistic system. In particular, Special Relativity on the Minkowski space $Q=\mathbb {R} ^{4}$ is of this type.

Since a configuration space $Q$ of a relativistic system has no preferable fibration over $\mathbb {R}$ , 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 $J_{1}^{1}Q\to Q$ 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 $J_{1}^{1}Q$ is provided with the adapted coordinates $(q^{0},q^{i},q_{0}^{i})$ possessing transition functions

q'^{0}=q'^{0}(q^{0},q^{k}),\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 $\mathbb {R} \times TQ$ , coordinated by $(\tau ,q^{\lambda },a_{\tau }^{\lambda })$ , where $TQ$ is the tangent bundle of $Q$ . Then a generic equation of motion of a relativistic system in terms of relativistic velocities reads

\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_{\alpha _{1}\ldots \alpha _{2N}}q_{\tau }^{\alpha _{1}}\cdots q_{\tau }^{\alpha _{2N}}=1.

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

References

Krasil'shchik, I. S., Vinogradov, A. M., [et al.], "Symmetries and conservation laws for differential equations of mathematical physics", Amer. Math. Soc., Providence, RI, 1999, ISBN 0-8218-0958-X.
Giachetta, G., Mangiarotti, L., Sardanashvily, G., Geometric Formulation of Classical and Quantum Mechanics (World Scientific, 2010) ISBN 981-4313-72-6 (arXiv: 1005.1212).

References

See also