Monge-gauge Hamiltonian of a three-dimensional string ($\mathcal{H}=\sqrt{(1+{\Phi }^{\prime 2})(1+{\Pi }^2)}$)

With unit tension and $X^0=t$, $X^1=\sigma$, the Virasoro constraints give $P_1=-\Pi {\Phi }^{\prime }$ and $P_0=-\sqrt{(1+{\Phi }^{\prime 2})(1+{\Pi }^2)}$. The reduced energy density is $\mathcal{H}=-P_0$. On the chiral branch $\Pi ={\Phi }^{\prime }$, the Hamiltonian field equation becomes $\dot{\Phi }={\Phi }^{\prime }$.