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In mathematics, a [[non-autonomous system (mathematics)| non-autonomous system]] of [[ordinary differential equation]]s is defined to be a dynamic equation on a smooth [[fiber bundle]] <math>Q\to \mathbb R</math> over <math>\mathbb R</math>. 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]] <math>Q</math> whose fibration over <math>\mathbb R</math> is not fixed. Such a system admits transformations of a coordinate <math>t</math> on <math>\mathbb R</math> depending on other coordinates on <math>Q</math>. Therefore, it is called the '''relativistic system'''. In particular, [[Special Relativity]] on the
[[Minkowski space]] <math>Q= \mathbb R^4</math> is of this type.

Since a configuration space <math>Q</math> of a relativistic system has no
preferable fibration over <math>\mathbb R</math>, a
velocity space of relativistic system is a first order jet
manifold <math>J^1_1Q</math> of one-dimensional submanifolds of <math>Q</math>. The notion of jets of submanifolds
generalizes that of [[jet (mathematics)|jets of sections]]
of fiber bundles which are utilized in [[covariant classical field theory]] and
[[non-autonomous mechanics]]. A first order jet bundle <math>J^1_1Q\to
Q</math> 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 <math>(q^0, q^i)</math> on <math>Q</math>, a first order jet manifold <math>J^1_1Q</math> is provided with the adapted coordinates <math>(q^0,q^i,q^i_0)</math>
possessing transition functions

: <math>q'^0=q'^0(q^0,q^k), \quad q'^i=q'^i(q^0,q^k), \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}.</math>

The relativistic velocities of a relativistic system are represented by
elements of a fibre bundle <math>\mathbb R\times TQ</math>, coordinated by <math>(\tau,q^\lambda,a^\lambda_\tau)</math>, where <math>TQ</math> is the tangent bundle of <math>Q</math>. Then a generic equation of motion of a relativistic system in terms of relativistic velocities reads

: <math> \left(\frac{\partial_\lambda G_{\mu\alpha_2\ldots\alpha_{2N}}}{2N}- \partial_\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,</math>

: <math>G_{\alpha_1\ldots\alpha_{2N}}q^{\alpha_1}_\tau\cdots q^{\alpha_{2N}}_\tau=1.</math>

For instance, if <math>Q</math> is the Minkowski space with a Minkowski metric <math>G_{\mu\nu}</math>, 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., [[Gennadi Sardanashvily|Sardanashvily, G.]], Geometric Formulation of Classical and Quantum Mechanics (World Scientific, 2010) ISBN 981-4313-72-6 ([http://xxx.lanl.gov/abs/1005.1212 arXiv: 1005.1212]).

==See also==
* [[Non-autonomous system (mathematics)]]
* [[Non-autonomous mechanics]]
* [[Relativistic mechanics]]
* [[Special relativity]]

[[Category:Differential equations]]
[[Category:Classical mechanics]]
[[Category:Theory of relativity]]

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