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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

$q{\text{'}}^0=q{\text{'}}^0(q^0,q^k),q{\text{'}}^i=q{\text{'}}^i(q^0,q^k),{q^{\prime }}_0^i=(\frac{\partial q{\text{'}}^i}{\partial q^j}{q}_0^j+\frac{\partial q{\text{'}}^i}{\partial q^0}){(\frac{\partial q{\text{'}}^0}{\partial q^j}{q}_0^j+\frac{\partial q{\text{'}}^0}{\partial q^0})}^{-1}.$

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

$(\frac{{\partial }_{\lambda }{G}_{\mu {\alpha }_2\dots {\alpha }_{2N}}}{2N}-{\partial }_{\mu }{G}_{\lambda {\alpha }_2\dots {\alpha }_{2N}})q_{\tau }^{\mu }{q}_{\tau }^{{\alpha }_2}\cdots q_{\tau }^{{\alpha }_{2N}}-(2N-1)G_{\lambda \mu {\alpha }_3\dots {\alpha }_{2N}}{q}_{\tau \tau }^{\mu }{q}_{\tau }^{{\alpha }_3}\cdots q_{\tau }^{{\alpha }_{2N}}+F_{\lambda \mu }{q}_{\tau }^{\mu }=0,$

$G_{{\alpha }_1\dots {\alpha }_{2N}}{q}_{\tau }^{{\alpha }_1}\cdots q_{\tau }^{{\alpha }_{2N}}=1.$

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).

See also

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