h_{AB}dx^A dx^B + (\tilde h^{zz}q^2 -1) N^2 d\sigma^2 =0.
\label{zeta}
\end{aligned}$$ Inserting and into , we obtain Jacobi’s form of the abbreviated action, $$\begin{aligned}
    S_0 &=\int \frac{1}{N}\tilde h_{AB} x'^A x'^B d\sigma \cr
        &= \int \sqrt{(1-\tilde h^{zz}q^2)\tilde h_{AB}dx^A dx^B}.\end{aligned}$$ Therefore, the action reduces to the geodesic action $$\begin{aligned}
    S_0 = \int \sqrt{h^{red}_{AB}dx^A dx^B}, 
\label{2dim-action}
\end{aligned}$$ where the metric of reduced two-dimensional space is given as $$\begin{aligned}
    h^{red}_{AB}= (1- \tilde h^{zz}q^2) \tilde h_{AB}.
\label{hred}
\end{aligned}$$

By variation of $S$ by $q$, we have $$\begin{aligned}
      z = -\frac{\partial S_0}{\partial q}.
\label{z}
\end{aligned}$$ The geodesic curves extremizing together with determine the full three-dimensional geodesics in the orbit space.

For the stationary rotating strings, the metric of two-dimensional curved space on which we seek geodesics is expressed explicitly in the form $$\begin{aligned}
    ds_{red}^2 &= h^{red}_{AB} dx^A dx^B 
        = (1- \tilde h^{zz}q^2) \tilde h_{AB} dx^A dx^B  \cr
        &= (1-q^2-\Omega^2 \rho^2)
        \left(d \rho^2 +\frac{ \rho^2}{1-\Omega^2 \rho^2}d  \varphi^2\right).
\label{2dim-metric}\end{aligned}$$ The action for geodesics is equivalent to $$\begin{aligned}
    S_0 &= \frac{1}{2}\int \left[ \frac{1}{N_{(2)}} (1-q^2-\Omega^2 \rho^2)
        \left( \rho'^2 +\frac{ \rho^2}{1-\Omega^2 \rho^2}\varphi'^2\right)
        + N_{(2)} \right]d\sigma, 
\label{2dim-P_action}\end{aligned}$$ where $N_{(2)}$ is an arbitrary function of $\sigma$ which is related with the freedom of parametrization on a curve.

By variation of the action by 