the first viewpoint, and also give brief discussion with the second view in Appendix A.

Action is written in the Hamilton form $$\begin{aligned}
    S = \int \left[p_i {  x}'^i 
            - N{\cal H} \right] d\sigma ,
\label{H_action}
\end{aligned}$$ where the canonical conjugate momentum $p_i$ with respect to $  x^i$ is $$\begin{aligned}
        p_i = \frac{1}{N}\tilde h_{ij}   x'^j.
\label{momentum}
\end{aligned}$$ Variation of by $N$ leads that the Hamiltonian $N{\cal H}$ is vanishing, i.e., $$\begin{aligned}
    {\cal H} = \frac{1}{2}\left( \tilde h^{ij} p_i p_j -1 \right)=0.
\label{Hconstraint}
\end{aligned}$$

Since the momentum $p_z$, which is conjugate to the cyclic coordinate $z$, takes a constant value, say $q$. The action reduces to $$\begin{aligned}
    S = S_0+ q   z - \int N{\cal H}  d\sigma , 
\end{aligned}$$ where $$\begin{aligned}
    S_0 = \int p_A x'^A  d\sigma, \qquad (x^A= \rho,  \varphi). 
\label{Sred}
\end{aligned}$$ When we concentrate on the curves that satisfy $p_z=q$ and ${\cal H}=0$, according to Maupertuis’ principle, the projected curve is the curve that extremizes the abbreviated action, $S_0$.

With the help of the Hamiltonian constraint is written in the form $$\begin{aligned}
            0={\cal H} &=\frac{1}{2}\left( \tilde h^{AB}p_A p_B 
        + \tilde h^{zz}q^2 -1 \right) \cr
                &=\frac{1}{2}\left(\frac{1}{N^2}\tilde h_{AB}  x'^A   x'^B 
                + \tilde h^{zz}q^2 -1 \right). \end{aligned}$$ Then, we have $$\begin{aligned}
     \tilde 