Parabolic.

The Elliptical cylinder.

The coordinate system primarily utilized in three-dimensional systems is the cylindrical coordinates of the system. The cylindrical coordinate system is used to find the surface area in three-dimensional space.

The velocity and scale factors are given below.

$$\begin{gathered} h_u=\sqrt{u^2+v^2} \\ {h}_v=\sqrt{u^2+v^2} \\ {h}_{\varphi }=uv \\ \dot{r}=\dot{u}\sqrt{u^2+v^2}{\stackrel{⏜}{e}}_u+\dot{v}\sqrt{u^2+v^2}{\stackrel{⏜}{e}}_v+\varphi uv{\stackrel{⏜}{e}}_{\varphi } \end{gathered}$$

The lagrangian in the parabolic coordinates are given below.

$$\begin{gathered} L=\frac{1}{2}m{\dot{r}}^2-V\left(u,v,\varphi \right) \\ L=\frac{1}{2}m\left[\left(u^2+v^2\right){\dot{u}}^2+\left(u^2+v^2\right){\dot{v}}^2+u^2{v}^2{\varphi }^2\right]-V\left(u,v,\varphi \right) \end{gathered}$$

Apply Euler-lagrange equation, the above equation becomes as follows.

$$\begin{gathered} \frac{d}{dt}\left(\frac{\partial L}{\partial \dot{u}}\right)=m\left[\left(u^2+v^2\right)\ddot{u}+u{\dot{u}}^2+2vv\dot{u}\right] \\ =m\left({\dot{v}}^2+{\dot{\varphi }}^2\right)u-\frac{\partial V}{\partial u} \end{gathered}$$

Solve further.

$$\begin{gathered} m\left[\ddot{u}\sqrt{u^2+v^2}+\frac{u}{\sqrt{u^2+v^2}}{\dot{u}}^2+\frac{2v}{\sqrt{u^2+v^2}}\dot{u}\dot{v}\right]+m\left[\frac{u}{\sqrt{u^2+v^2}}\dot{v}-\frac{u}{\sqrt{u^2+v^2}}{\dot{\varphi }}^2\right]=-\frac{1}{h_2}\frac{\partial V}{\partial u} \\ m\left[\ddot{v}\sqrt{u^2+v^2}-\frac{u}{\sqrt{u^2+v^2}}{\dot{u}}^2+\frac{2v}{\sqrt{u^2+v^2}}\dot{u}\dot{v}\right]+\left[\frac{v}{\sqrt{u^2+v^2}}{\dot{v}}^2-\frac{v}{\sqrt{u^2+v^2}}{\dot{\varphi }}^2\right]=-\frac{1}{h_v}\frac{\partial V}{\partial u} \\ m\left[\ddot{\varphi }uv+2v\dot{u}\dot{\varphi }+2v\dot{v}\dot{\varphi }\right]=-\frac{1}{h_2}\frac{\partial V}{\partial \varphi } \end{gathered}$$

Divide the above equation by respective scalar factors.

$$\begin{gathered} a_u=\ddot{u}\sqrt{u^2+v^2}+\frac{u}{\sqrt{u^2+v^2}}{\dot{u}}^2+\frac{2v}{\sqrt{u^2+v^2}}\dot{u}\dot{v}+\frac{u}{\sqrt{u^2+v^2}}\dot{v}-\frac{u}{\sqrt{u^2+v^2}}{\dot{\varphi }}^2 \\ {a}_v=\ddot{v}\sqrt{u^2+v^2}-\frac{u}{\sqrt{u^2+v^2}}{\dot{u}}^2+\frac{2u}{\sqrt{u^2+v^2}}\dot{u}\dot{v}+\frac{v}{\sqrt{u^2+v^2}}{\dot{v}}^2-\frac{v}{\sqrt{u^2+v^2}}{\dot{\varphi }}^2 \\ {a}_z=\ddot{\varphi }uv+2v\dot{u}\dot{\varphi }+2v\dot{v}\dot{\varphi } \end{gathered}$$

$$\begin{gathered} a_u=\ddot{u}\sqrt{u^2+v^2}+\frac{u}{\sqrt{u^2+v^2}}{\dot{u}}^2+\frac{2v}{\sqrt{u^2+v^2}}\dot{u}\dot{v}+\frac{u}{\sqrt{u^2+v^2}}{\dot{v}}^2 \\ {a}_v=\ddot{v}\sqrt{u^2+v^2}-\frac{u}{\sqrt{u^2+v^2}}{\dot{u}}^2+\frac{2u}{\sqrt{u^2+v^2}}\dot{u}\dot{v}+\frac{v}{\sqrt{u^2+v^2}}{\dot{v}}^2 \\ {a}_z=\ddot{z} \end{gathered}$$