= Solution
Let a dot denote differentiation with respect to the five-dimensional <affine parameter> $\lambda=\tilde u$, and define $W=\dot x^4+A_a\dot x^a$. Use the quadratic <geodesic Lagrangian>
$$
L=\frac12\left[\eta_{ab}\dot x^a\dot x^b+W^2\right].
$$
The assumed independence of $x^4$ makes it a <cyclic coordinate>, so the corresponding <Euler-Lagrange equation> is
$$
\frac{d}{d\lambda}\frac{\partial L}{\partial\dot x^4}=0.
$$
Since $\partial L/\partial\dot x^4=W$, this gives
$$
\boxed{\dot x^4+A_a\dot x^a=\xi,\qquad \dot\xi=0.}
$$
Geometrically, this is the conserved momentum associated with the fifth-direction <Killing vector field> $\partial_4$. It is the conserved fibre momentum underlying the <Kaluza-Klein geodesic Lorentz force>.
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