= Solution
Use the <Minkowski metric> $\eta_{mn}=\operatorname{diag}(-1,1,\ldots,1)$ and units with <speed of light> one. The contractions are $P^2=\eta^{mn}P_mP_n$ and $(TX')^2=T^2\eta_{mn}(X^m)'(X^n)'$; $(X^m)'P_m$ and $\dot X^mP_m$ pair a <vector> with a <covector> without another metric. The <Lagrange multipliers> impose the two <first-class constraints>
$$
\mathcal H=\tfrac12(P^2+T^2X'^2)=0,\qquad \mathcal D=X'\cdot P=0.
$$
They generate <worldsheet diffeomorphisms>, so the <phase space> contains both constrained directions and <gauge redundancy>. Two <first-class constraints> remove two <canonical pairs>, leaving $D-2$ physical <degrees of freedom> per point.
In <Monge gauge>, $X^0=t$ and $X^1=\sigma$. Write the transverse <canonical variables> as $\boldsymbol X=(X^2,\ldots,X^{D-1})$ and $\boldsymbol P=(P_2,\ldots,P_{D-1})$. Solving the <first-class constraints> gives
$$
P_1=-\boldsymbol X'\cdot\boldsymbol P,\qquad
P_0=-\mathcal E,\qquad
\mathcal E=\sqrt{T^2(1+|\boldsymbol X'|^2)+|\boldsymbol P|^2+(\boldsymbol X'\cdot\boldsymbol P)^2}.
$$
The negative root selects positive <energy>. Substitution into the <phase-space action> gives the <Hamiltonian> reduction
$$
\boxed{I_{\mathrm{red}}=\int dt\int_0^\pi d\sigma\,
\bigl(\dot{\boldsymbol X}\cdot\boldsymbol P-\mathcal E\bigr).}
$$
For a static segment, $\boldsymbol P=0$, its <proper length> element is $d\ell=\sqrt{1+|\boldsymbol X'|^2}\,d\sigma$, and $\mathcal E\,d\sigma=T\,d\ell$. Thus \b[the <string tension> is the <rest energy> per unit <proper length>]. In particular a straight resting segment has $\mathcal E=T$. <Monge gauge> is a local choice on a <string embedding map> for which $X^1$ is a valid coordinate; it need not cover folded strings or all endpoint configurations.
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