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

Since a configuration space ${\displaystyle Q}$ of a relativistic system has no preferable fibration over ${\displaystyle \mathbb {R} }$, a velocity space of relativistic system is a first order jet manifold ${\displaystyle J_{1}^{1}Q}$ of one-dimensional submanifolds of ${\displaystyle Q}$. 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 ${\displaystyle J_{1}^{1}Q\to Q}$ 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 ${\displaystyle (q^{0},q^{i})}$ on ${\displaystyle Q}$, a first order jet manifold ${\displaystyle J_{1}^{1}Q}$ is provided with the adapted coordinates ${\displaystyle (q^{0},q^{i},q_{0}^{i})}$ possessing transition functions

${\displaystyle q'^{0}=q'^{0}(q^{0},q^{k}),\quad q'^{i}=q'^{i}(q^{0},q^{k}),\quad {q'}_{0}^{i}=\left({\frac {\partial q'^{i}}{\partial q^{j}}}q_{0}^{j}+{\frac {\partial q'^{i}}{\partial q^{0}}}\right)\left({\frac {\partial q'^{0}}{\partial q^{j}}}q_{0}^{j}+{\frac {\partial q'^{0}}{\partial q^{0}}}\right)^{-1}.}$

The relativistic velocities of a relativistic system are represented by elements of a fibre bundle ${\displaystyle \mathbb {R} \times TQ}$, coordinated by ${\displaystyle (\tau ,q^{\lambda },a_{\tau }^{\lambda })}$, where ${\displaystyle TQ}$ is the tangent bundle of ${\displaystyle Q}$. Then a generic equation of motion of a relativistic system in terms of relativistic velocities reads

${\displaystyle \left({\frac {\partial _{\lambda }G_{\mu \alpha _{2}\ldots \alpha _{2N}}}{2N}}-\partial _{\mu }G_{\lambda \alpha _{2}\ldots \alpha _{2N}}\right)q_{\tau }^{\mu }q_{\tau }^{\alpha _{2}}\cdots q_{\tau }^{\alpha _{2N}}-(2N-1)G_{\lambda \mu \alpha _{3}\ldots \alpha _{2N}}q_{\tau \tau }^{\mu }q_{\tau }^{\alpha _{3}}\cdots q_{\tau }^{\alpha _{2N}}+F_{\lambda \mu }q_{\tau }^{\mu }=0,}$

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

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

See also

• Non-autonomous system (mathematics)
• Non-autonomous mechanics
• Relativistic mechanics
• Special relativity

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).