= Kaluza-Klein geodesic Lorentz force
{c}
{title2=$\frac{D\dot x^i}{ds}=kF^i{}_j\dot x^j$}
For $ds_5^2=\gamma_{ij}dx^idx^j+(dz+A_idx^i)^2$ with $z$-independent coefficients, the <geodesic Lagrangian> makes $k=\dot z+A_i\dot x^i$ constant. Its remaining <Euler-Lagrange equations> give the displayed <Lorentz force>, where $F=dA$ and $D$ uses the <Levi-Civita connection> of $\gamma$. For a particle of five-dimensional mass $M$ with proper-time parameter $s$, the conserved momentum is $p_z=Mk$ and the four-dimensional rest mass is $m=M\sqrt{1+k^2}$. Reparametrizing by four-dimensional proper time gives $Du^i/d\tau=(p_z/m)F^i{}_ju^j$.
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