\\
    P &:= \int_{\rho_{min}}^{\rho_{max}} d\bar\rho\int_0^{2 \pi} d\bar\varphi 
            \int_0^{Z_p} d\bar z 
            \sqrt{-g}~T^{\bar t}_{\ \bar\nu} (\partial_{\bar z})^{\bar \nu}
        = -\mu\int_0^{n_l\sigma_p}d\sigma\ 
                \Theta^{\bar t}_{~\bar z}(\sigma).\end{aligned}$$ It is easy to calculate these quantities as $$\begin{aligned}
    E &= \frac{\pi \mu }{|\Omega|} n_\ell (1-\ell^2), 
\label{EDef} \\
    J &= \frac{\pi \mu }{2\Omega |\Omega|} n_\ell (1-\ell^2 -q^2), 
\label{JDef} \\
    P &= - \frac{\pi\mu}{\Omega} n_\ell  \ell q .
\label{PDef}   \end{aligned}$$ We can also define the averaged values of theses quantities per unit length of $z$ as $$\begin{aligned}
    \braket{E} &:= E/Z_p 
        = \mu \frac{ 1-\ell^2 }{|q|} , 
\\
    \braket{J} &:= J/Z_p 
        =\frac{\mu}{\Omega} \frac{ 1-\ell^2 -q^2 }{2|q|} , 
\label{hEJPDef} \\
    \braket{P} &:= P/Z_p 
        =-\mu \ell \mathrm{sign}(\Omega q).\end{aligned}$$ These quantities are applicable for the strings with irrational $\ell$.

When $\ell\neq 0$ and $q\neq 0$, we see $P\neq 0 $, that is, the strings move along the rotating axis. Although the rotating velocity of a string segment is perpendicular to the rotation axis, the physical velocity of the Nambu-Goto string is orthogonal to the segment. Since the strings with $\ell\neq 0$ and $q\neq 0$ have inclination to the rotation axis, then the physical velocity of string segments have the component along the rotation axis. This is the reason for non-vanishing $P$. In the planar 