# Solving the equation of relativistic motion

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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With a purely magnetic field (no E field) and the given initial conditions, the force vanishes and the solution is trivial. – Ben Crowell May 9 '13 at 1:23
There is actually more than one equation, or an equation with a constraint. You should take into account a $p_ap^b=m^2$ constraint, or likewise a $d(p_ap^b)/d\tau=0$ equation. And tensor equations are systems of real equations if you write down all tensors components of an equation. – firtree May 9 '13 at 4:13
@BenCrowell: Thanks!;) – a very confused person May 9 '13 at 8:36
@BenCrowell: What about if the initial condition was $p^a=(E_0, \vec p_0)$? Can the conditions of orthogonality reduce the problem i some way? i think it allows me to rotate the coordinates so as to reduce the number of on-vanishing tensor components in the field tensor, but can it do anything else? – a very confused person May 9 '13 at 8:59