D4a - motion of charged particles

the Lorentz force:

F^{μ} = q F^{μ}_{ν} u^{ν}

applying Newton's second law:

\frac{d u^{μ}}{d τ} = \frac{q}{m} F^{μ}_{ν} u^{ν}

solve it as an eigenvalue equation as:

\frac{d \vec{u}}{d τ} = M \vec{u}

the general solution is:

\vec{u} = \sum_{λ} A_{λ} e^{λ t} {\vec{e}}_{λ}

where, $λ$ are the eigen values, $A_{λ}$ is the amplitude, and ${\vec{e}}_{λ}$ is the eigenvector of $M$

to find $A_{λ}$ , use the initial conditions
considering some special cases