Affine connection for a velocity-linear force (${\ddot{x}}^i=E^i+C^i{}_j{\dot{x}}^j$)

For smooth spatial fields $E^i,C^i{}_j$, define a torsion-free connection on $\mathbb{R}\times {\mathbb{R}}^n$ by ${\Gamma }^i{}_{00}=-E^i$, ${\Gamma }^i{}_{0j}={\Gamma }^i{}_{j0}=-C^i{}_j/2$ and all other coefficients zero. The time geodesic equation is $\ddot{t}=0$. For $\dot{t}\ne 0$, use $t$ as affine parameter; the spatial geodesic equation becomes $d^2{x}^i/d{t}^2=E^i+C^i{}_{j}d{x}^j/dt$. In dimension three, $C^i{}_j=2{ϵ}^i{}_{kj}{B}^k$ realizes $\ddot{\mathbf{x}}=\mathbf{E}+2\mathbf{B}\times \dot{\mathbf{x}}$. Such a connection need not preserve a metric; a spacetime description of these trajectories only needs affine geometry.