Relativistic system (mathematics) explained

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

Since a configuration space

Q of a relativistic system has nopreferable fibration over R , a velocity space of relativistic system is a first order jetmanifold J ^{1} _{1 Q} of one-dimensional submanifolds of Q . The notion of jets of submanifoldsgeneralizes that of jets of sectionsof fiber bundles which are utilized in covariant classical field theory andnon-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 spacesof 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} ^{i} ^{0 )} possessing transition functions q' ^{0=q'} ^{0(q} ^{0,q} ^{k),} q' ^{i=q'} ^{i(q} ^{0,q} ^{k),} {q'} ^{i} _{0} = \left( (\partial q' ^{i})/(\partial q ^{j}) q ^{j} _{0} + (\partial q' ^{i})/(\partial q ^{0}) \right) \left( (\partial q' ^{0})/(\partial q ^{j}) q ^{j} _{0} + (\partial q' ^{0})/(\partial q ^{0}) \right) ^{-1} .

The relativistic velocities of a relativistic system are represented byelements of a fibre bundle

R x TQ , coordinated by (\tau,q ^{λ,a} ^{λ} ^{\tau)} , 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( (\partial _{λ} G _{\mu\alpha _{2\ldots\alpha} _{2N}})/(2 N) - \partial _{\mu G} ^{λ \alpha _{2\ldots\alpha} _{2N}} \right) q ^{\mu} _{\tau} q ^{\alpha _{2N}} _{\tau} - ( 2 N -1 )G _{λ \mu\alpha _{3\ldots\alpha} _{2N}} q ^{\mu} _{\tau\tau} q ^{\alpha _{2N}} _{\tau} + F _{λ\mu} q ^{\mu} _{\tau} =0 , G _{\alpha _{1\ldots\alpha} _{2N}} q ^{\alpha _{1}} _{\tau …} q ^{\alpha _{2N}} _{\tau =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.

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, .
• Giachetta, G., Mangiarotti, L., Sardanashvily, G., Geometric Formulation of Classical and Quantum Mechanics (World Scientific, 2010) .