= Affine connection for a velocity-linear force
{title2=$\ddot x^i=E^i+C^i{}_j\dot x^j$}
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 $\dot t\ne0$, use $t$ as <affine parameter>; the spatial geodesic equation becomes $d^2x^i/dt^2=E^i+C^i{}_jdx^j/dt$. In dimension three, $C^i{}_j=2\epsilon^i{}_{kj}B^k$ 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.
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