= Solution
Differentiate the <geodesic Lagrangian> before substituting the conserved quantity. For a four-dimensional index $a$,
$$
\frac{\partial L}{\partial\dot x^a}=\eta_{ab}\dot x^b+WA_a,
\qquad
\frac{\partial L}{\partial x^a}=W(\partial_aA_c)\dot x^c.
$$
Its <Euler-Lagrange equation>, together with $W=\xi$ and $\dot W=0$, is
$$
\eta_{ab}\ddot x^b+\xi(\partial_cA_a)\dot x^c
-\xi(\partial_aA_c)\dot x^c=0.
$$
Thus the <sign of the projected Kaluza-Klein Lorentz force> is fixed by the printed metric:
$$
\boxed{\ddot x^a=\xi\eta^{ab}(\partial_bA_c-\partial_cA_b)\dot x^c
=\xi F^a{}_c\dot x^c.}
$$
\b[The derivative order in the PDF's proposed equation is reversed.] Its right-hand side is the negative of the result above for the stated $\xi$ and metric. This is not repaired by renaming the field tensor, because the two explicit derivative expressions have opposite signs. For example, $A_0=A_0(x^1)$, $A_1=0$ gives $\ddot x^1=-\xi A_0'(x^1)\dot x^0$ directly from the Lagrangian, whereas the printed expression gives the opposite. The corrected equation is used below.
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