Relativistic system (mathematics)

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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).

Relativistic system (mathematics)

References

See also