= Solution
For the <Monge-gauge Hamiltonian of a three-dimensional string>, in <Monge gauge>, $X'^m=(0,1,\Phi')$ and $P_m=(P_0,P_1,\Pi)$. The <momentum> <constraint> gives $P_1=-\Phi'\Pi$. The other <Virasoro constraint> then reads
$$
-P_0^2+(\Phi'\Pi)^2+\Pi^2+1+\Phi'^2=0.
$$
Choosing the positive-energy branch,
$$
\boxed{P_1=-\Phi'\Pi,\qquad P_0=-\sqrt{(1+\Phi'^2)(1+\Pi^2)}.}
$$
Substituting into the <phase-space action> leaves $I=\int dt\,d\sigma(\Pi\dot\Phi-\mathcal H)$, so
$$
\boxed{F(z)=1+z^2,\qquad \mathcal H=\sqrt{(1+\Phi'^2)(1+\Pi^2)},\qquad H=\int d\sigma\,\mathcal H.}
$$
For an infinite string this is an energy-density expression; even the undeformed string has infinite total <energy>. One may regulate the length or subtract its constant reference <energy> when a finite total <energy> is needed. The positive branch is unambiguous at the density level.
The canonical <Hamiltonian field equations> are
$$
\dot\Phi=\Pi\sqrt{\frac{1+\Phi'^2}{1+\Pi^2}},\qquad
\dot\Pi=\partial_\sigma\left(\Phi'\sqrt{\frac{1+\Pi^2}{1+\Phi'^2}}\right).
$$
Consequently $\Pi=\Phi'$ implies
$$
\boxed{\dot\Phi=\Phi',\qquad\Phi=f(\sigma+t).}
$$
The second equation also becomes $\dot\Pi=\Phi''$, consistent with differentiating the first. These are one-direction traveling profiles along the string.
For the linear profile, the spatial string is the straight line $X^2=kX^1+kt$. It is tilted, and its coordinate intercept moves with speed $k$. This is also a uniformly moving straight string: the <physical transverse velocity of a tilted string> is the velocity normal to the line. With unit tangent $(1,k)/\sqrt{1+k^2}$ and coordinate velocity $(0,k)$, the normal speed is
$$
\boxed{v_\perp=\frac{|k|}{\sqrt{1+k^2}}<1.}
$$
Thus \b[$k>1$ does not imply superluminal physical motion]. Fixed-$\sigma$ labels have spacelike trajectories when $k>1$, but labels on a string can slide tangentially and are not identifiable material particles. Choosing $\dot\sigma=-k^2/(1+k^2)$ removes the tangential velocity and gives the subluminal velocity $(-k^2,k)/(1+k^2)$.
The <induced worldsheet metric> supplies a coordinate-invariant check:
$$
h_{tt}=k^2-1,\qquad h_{t\sigma}=k^2,\qquad h_{\sigma\sigma}=1+k^2,\qquad\boxed{\det h=-1.}
$$
The <string worldsheet> remains Lorentzian for every finite $k$, including $k=1$ and $k>1$. Its constant <energy density> is $1+k^2$. This linear profile is a tilted boosted straight string, rather than a localized finite-energy wave.
\Image[/past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2016/iii/paper-306-transverse-motion.png]
{title=For slope k=2 the vertical coordinate velocity exceeds one, while its component normal to the tilted string has physical speed 2 divided by the square root of 5}
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