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Essentially, How does one solve the tensorial differential equation $$\frac{dx^a}{d\tau}=A^a{}_bx^b$$

where $x^a$ is a 4-vector and $A^a{}_b$ is a $(1,1)$ tensor.


The original Problem

How does one solve the tensor differential equation for the relativistic motion of a partilcle of charge $e$ and mass $m$, with 4-momentum $p^a$ and electromagnetic field tensor $F_{ab}$ of a constant magetic field $\vec B$ perpendicular to the plane of motion. $$\frac{dp^a}{d\tau}=\frac{e}{m}F^a{}_bp^b$$ ? Let the the initial condition be $$p^a=(E_0 ,\vec 0)$$

I can see that the differential equation resembles that of a SHM equation or a cosh, sinh one if it's a scalar equation. However, I don't know how to deal with a tensor equation. Could anyone please explain? Thank you.

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  • $\begingroup$ What if you replace tensor by matrix ? $\endgroup$ – Damien L May 9 '13 at 7:56
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Whenever I have come across differential equations involving tensors normally I had written them in component form and then resorted to normal methods of solving systems of differential equations...although there are a few tensorial differential equations that can be solved not by that method, like the Dirac equation

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  • $\begingroup$ Also I found a pdf materias.fi.uba.ar/6760/sokolnikoff.pdf, wherein after page 209 they start discussing problems in physics in terms of tensors. $\endgroup$ – Triatticus May 9 '13 at 1:08
  • $\begingroup$ I should come up with some examples for each. For example Einsteins Field Equations when using simplifying assumptions (like maximal spherical symmetry, and static for Schwarzschild metric) the tensor equation lends itself more easily to solution for the metric via component representation. In field theory I have seen the Dirac and Klein-Gordon fields delt with through differeing means like fourier integrals. $\endgroup$ – Triatticus May 17 '13 at 22:06

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