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 over . 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 whose fibration over is not fixed. Such a system admits transformations of a coordinate on depending on other coordinates on . Therefore, it is called the relativistic system. In particular, Special Relativity on the Minkowski space is of this type.

Since a configuration space of a relativistic system has no preferable fibration over , a velocity space of relativistic system is a first order jet manifold of one-dimensional submanifolds of . 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 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 on , a first order jet manifold is provided with the adapted coordinates possessing transition functions

The relativistic velocities of a relativistic system are represented by elements of a fibre bundle , coordinated by , where is the tangent bundle of . Then a generic equation of motion of a relativistic system in terms of relativistic velocities reads

For instance, if is the Minkowski space with a Minkowski metric , this is an equation of a relativistic charge in the presence of an electromagnetic field.

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References[edit]

  • 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).