= Solution
In <Bohmian mechanics> the particle has a definite position $\mathbf X(t)$ at every time. Its <wavefunction> obeys the usual autonomous wave equation, while its actual position follows the <guidance equation>
$$
\dot{\mathbf X}(t)=\mathbf v(\mathbf X(t),t),\qquad
\mathbf v=\frac{\hbar}{m}\nabla S=\frac{\mathbf j}{|\psi|^2}.
$$
Here $\mathbf j$ is the <probability current>. The <Born rule> is the quantum-equilibrium choice of initial position distribution $\rho=|\psi|^2$; <quantum equilibrium equivariance> ensures that this distribution persists because it obeys the same <probability continuity equation> as the wave amplitude. The <guidance equation> fixes the initial velocity as well as subsequent velocities: the second-order equation below does not permit an independent arbitrary initial velocity.
Define the <quantum potential>
$$
\boxed{Q=-\frac{\hbar^2}{2m}\frac{\nabla^2R}{R}}.
$$
Taking the <gradient> of the real <Madelung equations> gives
$$
\hbar\partial_t\nabla S+\nabla\left(\frac{\hbar^2}{2m}|\nabla S|^2\right)=-\nabla(V+Q).
$$
On any smooth phase patch $\nabla\times\mathbf v=0$, so $\nabla(|\mathbf v|^2/2)=(\mathbf v\cdot\nabla)\mathbf v$. Along the actual path, differentiation is the <material derivative> $D/Dt=\partial_t+\mathbf v\cdot\nabla$. Therefore
$$
\boxed{\frac{d(m\dot{\mathbf X})}{dt}=m\frac{D\mathbf v}{Dt}=-\nabla(V+Q)\big|_{\mathbf X(t)}}.
$$
This is the <Bohmian mechanics> Newton form: the classical force is supplemented by the amplitude-dependent <quantum potential>. Neither division by $R$ nor a smooth phase is justified at a <wavefunction> node, so the derivation applies on nonzero-amplitude regions. A nonzero circulation around a node is compatible with the locally curl-free <guidance equation>.
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