= Sign of the projected Kaluza-Klein Lorentz force
Take $L=[\eta_{ab}\dot x^a\dot x^b+W^2]/2$ with $W=\dot y+A_a\dot x^a=\xi$ constant. The <Euler-Lagrange equation> for $x^a$ is
$$
\eta_{ab}\ddot x^b+\xi(\partial_cA_a-\partial_aA_c)\dot x^c=0.
$$
Thus, with $F_{bc}=\partial_bA_c-\partial_cA_b$,
$$
\ddot x^a=\xi\eta^{ab}(\partial_bA_c-\partial_cA_b)\dot x^c=\xi F^a{}_c\dot x^c.
$$
Reversing the explicit derivative order while retaining this metric and the same definition of $\xi$ changes the equation rather than merely changing a field-tensor naming convention. More generally, a fibre term $\sigma W^2$, $\sigma=\pm1$, gives $\sigma\xi F^a{}_b\dot x^b$; its conserved canonical fibre momentum is $p_y=\sigma\xi$.
In the normalized positive-fibre case with $|\xi|<1$, four-dimensional <proper time> gives $du^a/d\tau=c\xi F^a{}_bu^b/\sqrt{1-\xi^2}$. This has the <Lorentz force> form. The scale relating the geometrical $A$ to a physical <electromagnetic four-potential> fixes the conversion to an SI charge-to-mass ratio.
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