= Solution
Define the <open-string endpoint momentum flux> along the <open string> by $J_m=eT^2X'_m+uP_m$. Variation of the <Nambu-Goto phase-space action> gives
$$
\boxed{\dot X^m=eP^m+uX'^m,\qquad\dot P_m=\partial_\sigma J_m,}
$$
$$
\boxed{P^2+T^2X'^2=0,\qquad X'\cdot P=0.}
$$
The two last equations are the <Virasoro constraints>. Integration of the <momentum> equation gives
$$
\boxed{\dot{\mathcal P}_m=J_m(t,\pi)-J_m(t,0).}
$$
Thus the necessary and sufficient condition for total <momentum conservation> is equality of the two endpoint fluxes, component by component. Vanishing of both fluxes is a sufficient local boundary condition.
The spatial boundary term in the action variation is $-\int dt[J_m\delta X^m]_0^\pi$. For free ends the endpoint variations are arbitrary, so
$$
\boxed{J_m|_{0,\pi}=0\quad\Longrightarrow\quad\dot{\mathcal P}_m=0.}
$$
In a gauge with $u=0$ and nonzero $e$ this reduces to the usual <Neumann boundary condition> $X'^m=0$. The conclusion is conservation of total <momentum>, not that total <momentum> must vanish. The PDF contains the derivative in this conclusion; the TeX transcription drops it.
For endpoints confined to $x^1=0$, the <Dirichlet boundary condition> fixes $\delta X^1=0$. It therefore imposes no requirement that $J_1$ vanish. Preserving the fixed position requires $\dot X^1=eP^1+uX'^1=0$ at each endpoint, but that is a different condition. For example, in $u=0$ gauge the endpoint <momentum> $P^1$ is zero while $eT^2X'^1$ can be nonzero. Consequently \b[$\mathcal P_1$ is generally not conserved]: the normal endpoint forces transfer <momentum> to the hyperplane. A dynamical <D-brane> recoils and carries the missing <momentum>. If the hyperplane is idealized as fixed, its external support absorbs the <momentum>. The combined string-and-support system conserves <momentum>; freely varying directions tangent to the hyperplane retain their vanishing-flux condition.
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