= Solution
\b[Actions and equations.] Use a mostly-plus <Minkowski metric> and define the <induced worldsheet metric> $h_{ab}=\partial_aX\cdot\partial_bX$. For a nondegenerate timelike <worldsheet>, the <Nambu–Goto action> is
$$
S_{\mathrm{NG}}=-T\int d^2\sigma\sqrt{-\det h}.
$$
Varying the independent metric in the <Polyakov action> sets its <worldsheet stress tensor> to zero:
$$
h_{ab}-\frac12\gamma_{ab}\gamma^{cd}h_{cd}=0.
$$
In two dimensions this says $h_{ab}=\Omega^2\gamma_{ab}$ for a positive local factor. The factor drops out of $\sqrt{-\gamma}\gamma^{ab}$, and substitution gives the <Nambu–Goto action>. Conversely, any nondegenerate induced metric solves the auxiliary-metric equation up to a <Weyl transformation>. This <classical equivalence of Polyakov and Nambu–Goto actions> is a statement about classical embeddings; quantum equivalence additionally requires treatment of the metric measure and anomaly.
Variation of $X^m$ in the metric action, followed by elimination of $\gamma$, gives the <Nambu–Goto equations of motion>
$$
\boxed{\partial_a\!\left(\sqrt{-h}\,h^{ab}\partial_bX^m\right)=0.}
$$
A closed string has periodic $X$ and no spatial endpoint variation.
For a background metric $G$, <Kalb–Ramond field> $b$, and <dilaton> $\Phi$, one consistent Lorentzian convention is
$$
S_L=-\frac1{4\pi\alpha'}\int d^2\sigma\sqrt{-\gamma}\,
\gamma^{ab}G_{mn}(X)\partial_aX^m\partial_bX^n
+\frac1{4\pi\alpha'}\int d^2\sigma\,\epsilon^{ab}b_{mn}(X)\partial_aX^m\partial_bX^n
-\frac1{4\pi}\int d^2\sigma\sqrt{-\gamma}\,\Phi(X)R^{(2)},
\qquad T=\frac1{2\pi\alpha'}.
$$
The orientation fixes the two-form sign; here the Lorentzian curvature convention is chosen so Wick rotation gives the positive Euclidean dilaton term $S_{E,\Phi}=\int\sqrt\gamma\Phi R^{(2)}/(4\pi)$. This avoids hiding the convention in the topology argument. For constant vacuum value $\Phi_0$, the <Gauss-Bonnet theorem> gives $S_{E,\Phi}=\Phi_0\chi$. A connected closed oriented surface of genus $g$ has <Euler characteristic> $\chi=2-2g$, so
$$
\boxed{e^{-S_{E,\Phi}}=e^{-\Phi_0(2-2g)}
=g_s^{2g-2},\qquad g_s=e^{\Phi_0}.}
$$
The connected vacuum amplitude has a <string genus expansion> $\sum_{g\ge0}g_s^{2g-2}\mathcal A_g$; disconnected vacuum diagrams exponentiate the connected sum. Each additional handle supplies a factor $g_s^2$. This is <dilaton Euler-characteristic weighting>.
\b[The rotating circle.] For the specified embedding, the squared spatial speed and tangent length are both $1/2$, and their spatial inner product is zero. Including $X^0=t$ therefore gives
$$
\boxed{h_{ab}=\begin{pmatrix}-1/2&0\\0&1/2\end{pmatrix},\qquad
\sqrt{-h}=1/2.}
$$
The metric is constant. Its equation reduces to $\ddot X-X''=0$, satisfied by the left- and right-moving trigonometric components and the linear time component. On a constant-time slice with $0\le\sigma<2\pi$, the <proper length of a string> is
$$
\boxed{L=\int_0^{2\pi}\sqrt{h_{\sigma\sigma}}\,d\sigma
=\sqrt2\pi.}
$$
It is time-independent.
Let $R(\theta)$ be an ordinary two-dimensional rotation matrix. Rotate the first coordinate pair by $R(t)$ and the second by $R(-t)$. In these time-dependent Cartesian coordinates both pairs become $(\cos\sigma,\sin\sigma)/2$. A further fixed orthogonal change of basis gives
$$
Y_1=\frac{X_{\mathrm{rot}}^1+X_{\mathrm{rot}}^3}{\sqrt2}
=\frac{\cos\sigma}{\sqrt2},\qquad
Y_2=\frac{X_{\mathrm{rot}}^2+X_{\mathrm{rot}}^4}{\sqrt2}
=\frac{\sin\sigma}{\sqrt2},\qquad Y_3=Y_4=0.
$$
Thus every spatial slice is a planar circle of radius $1/\sqrt2$; its plane rotates in the ambient four-space. This is a <rigid circular string in four spatial dimensions>, not a pulsating circle. The rotating axes establish its spatial shape, not a transformation to an inertial spacetime frame.
The momentum density obtained from the <Nambu–Goto action> is $P_m=-T\sqrt{-h}\,h^{ta}\partial_aX_m$. Here $P_0=-T$, hence the conserved target-space energy is
$$
\boxed{E=-\int_0^{2\pi}P_0\,d\sigma=2\pi T=\sqrt2\,TL.}
$$
The material velocity is transverse to the tangent and has magnitude $1/\sqrt2$. Its Lorentz factor is $\sqrt2$, so the excess over $TL$ is kinetic energy. The spatial circle being stationary in rotating axes does not eliminate this energy.
\b[<Constraints> and endpoints.] Varying the multipliers in the <Nambu-Goto phase-space action> imposes
$$
C=\frac12(P^2+T^2X'^2)=0,\qquad D=P\cdot X'=0.
$$
These are <first-class constraints> generating normal and tangential <worldsheet diffeomorphisms>. They remove the two longitudinal embedding degrees of freedom; they are not extra physical force laws. Hamilton's equations are
$$
\boxed{\dot X^m=eP^m+uX'^m,\qquad
\dot P_m=(eT^2X'_m+uP_m)'.}
$$
For an open string, integration by parts leaves the spatial boundary variation
$$
\delta S_{\mathrm{end}}=-\int dt\,
\left[(eT^2X'_m+uP_m)\delta X^m\right]_{\sigma=0}^{\sigma=\pi}.
$$
Allowed <variational open-string boundary conditions> must make this vanish for every permitted endpoint displacement, and be preserved by the evolution. The bracketed expression is the <open-string endpoint momentum flux>. For free displacements, set that flux to zero. In a boundary-preserving gauge with $u=0$ and finite nonzero $e$, this is the <Neumann boundary condition> $X'^m=0$ at each end. This shows that <free-end string boundary conditions> are consistent.
At such an end, $C=0$ gives $P^2=0$, and $\dot X=eP$ gives $\dot X^2=0$. For a nontrivial endpoint trajectory with nonzero time velocity, its spatial speed is therefore one. This <null motion of a free string endpoint> follows from the boundary condition together with the <constraints>, not merely from the bulk wave equation. The induced metric may degenerate right at a free end; the result is understood as the endpoint limit or in the Polyakov description.
For a fixed spatial $p$-plane, split coordinates into $X^A$ along its $(p+1)$-dimensional worldvolume, including time, and $X^i$ transverse to it. Set $X'^A=0$ and $X^i=y^i$ at the ends. Tangential variations are free and their momentum flux vanishes; normal variations vanish, so their flux need not vanish. Evolution preserves the fixed normal values with $P^i=0$ at the boundary in the same gauge. These mixed <Neumann boundary conditions> and <Dirichlet boundary conditions> consistently restrict the endpoint worldlines to the plane.
The massless open-string excitations then separate into a gauge vector along the worldvolume and transverse <scalar fields>. A scalar displacement changes the corresponding $y^i$; its interpretation is a fluctuation of the plane's embedding. These <worldvolume fields from open-string massless states> support the interpretation as a dynamical planar <D-brane>. In the bosonic theory the still lower ground state is a <tachyon>, indicating an unstable brane; the interpretation does not require pretending that this <tachyon> is a stable massless field. Appropriate superstring sectors can remove that instability.
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