D4c - uniform magnetic field

\vec{E} = (\begin{matrix} 0 & 0 & 0 \end{matrix}) \vec{B} = (\begin{matrix} B_{x} & 0 & 0 \end{matrix})

F^{α}_{β} = (\begin{matrix} 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & B_{x} \\ 0 & 0 & - B_{x} & 0 \end{matrix})

\frac{d u^{α}}{d τ} = \frac{q}{m} F^{α}_{β} u^{β}

\frac{d u^{0}}{d τ} = 0 ⟹ u^{0} = γ c = constant

\frac{d u^{1}}{d τ} = 0 ⟹ u^{1} = γ v_{x} = constant

\frac{d}{d τ} (\begin{matrix} u^{2} \\ u^{3} \end{matrix}) = \frac{q}{m} B_{x} (\begin{matrix} 0 & 1 \\ - 1 & 0 \end{matrix}) (\begin{matrix} u^{2} \\ u^{3} \end{matrix})

u^{2} = u_{0}^{2} e^{- λ τ} λ_{1} = i, λ_{2} = - i (\begin{matrix} u^{2} \\ u^{3} \end{matrix}) = A (\begin{matrix} \sin (ω τ + ϕ) \\ \cos (ω τ + ϕ) \end{matrix})

where, $ω = q B_{x} / m$ is the cyclotron/Larmour frequency

d t = γ d τ ω (t) = \frac{1}{γ} ω (t) = \frac{q B_{x}}{m γ}