91Ó°ÊÓ

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.

$$\begin{gathered} h_u=\sqrt{u^2+v^2} \\ {h}_v=\sqrt{u^2+v^2} \\ {h}_{\mathrm{Ï•}}=uv \\ \stackrel{Ë™}{r}=\stackrel{Ë™}{u}\sqrt{u^2+v^2}{\stackrel{Áåœ}{e}}_u+\stackrel{Ë™}{v}\sqrt{u^2+v^2}{\stackrel{Áåœ}{e}}_v+\mathrm{Ï•}uv{\stackrel{Áåœ}{e}}_{\mathrm{Ï•}} \end{gathered}$$

The lagrangian in the parabolic coordinates are given below.

$$\begin{gathered} L=\frac{1}{2}m{\stackrel{Ë™}{r}}^2-V\left(u,v,\mathrm{Ï•}\right) \\ L=\frac{1}{2}m\left[\left(u^2+v^2\right){\stackrel{Ë™}{u}}^2+\left(u^2+v^2\right){\stackrel{Ë™}{v}}^2+u^2{v}^2{\mathrm{Ï•}}^2\right]-V\left(u,v,\mathrm{Ï•}\right) \end{gathered}$$

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

$$\begin{gathered} \frac{d}{dt}\left(\frac{∂L}{∂\stackrel{˙}{u}}\right)=m\left[\left(u^2+v^2\right)\stackrel{¨}{u}+u{\stackrel{˙}{u}}^2+2vv\stackrel{˙}{u}\right] \\ =m\left({\stackrel{˙}{v}}^2+{\stackrel{˙}{\mathrm{ϕ}}}^2\right)u-\frac{∂V}{∂u} \end{gathered}$$

Solve further.

$$\begin{gathered} m\left[\stackrel{¨}{u}\sqrt{u^2+v^2}+\frac{u}{\sqrt{u^2+v^2}}{\stackrel{˙}{u}}^2+\frac{2v}{\sqrt{u^2+v^2}}\stackrel{˙}{u}\stackrel{˙}{v}\right]+m\left[\frac{u}{\sqrt{u^2+v^2}}\stackrel{˙}{v}-\frac{u}{\sqrt{u^2+v^2}}{\stackrel{˙}{\mathrm{ϕ}}}^2\right]=-\frac{1}{h_2}\frac{∂V}{∂u} \\ m\left[\stackrel{¨}{v}\sqrt{u^2+v^2}-\frac{u}{\sqrt{u^2+v^2}}{\stackrel{˙}{u}}^2+\frac{2v}{\sqrt{u^2+v^2}}\stackrel{˙}{u}\stackrel{˙}{v}\right]+\left[\frac{v}{\sqrt{u^2+v^2}}{\stackrel{˙}{v}}^2-\frac{v}{\sqrt{u^2+v^2}}{\stackrel{˙}{\mathrm{ϕ}}}^2\right]=-\frac{1}{h_v}\frac{∂V}{∂u} \\ m\left[\stackrel{¨}{\mathrm{ϕ}}uv+2v\stackrel{˙}{u}\stackrel{˙}{\mathrm{ϕ}}+2v\stackrel{˙}{v}\stackrel{˙}{\mathrm{ϕ}}\right]=-\frac{1}{h_2}\frac{∂V}{∂\mathrm{ϕ}} \end{gathered}$$

Divide the above equation by respective scalar factors.

$$\begin{gathered} a_u=\stackrel{¨}{u}\sqrt{u^2+v^2}+\frac{u}{\sqrt{u^2+v^2}}{\stackrel{˙}{u}}^2+\frac{2v}{\sqrt{u^2+v^2}}\stackrel{˙}{u}\stackrel{˙}{v}+\frac{u}{\sqrt{u^2+v^2}}\stackrel{˙}{v}-\frac{u}{\sqrt{u^2+v^2}}{\stackrel{˙}{\mathrm{ϕ}}}^2 \\ {a}_v=\stackrel{¨}{v}\sqrt{u^2+v^2}-\frac{u}{\sqrt{u^2+v^2}}{\stackrel{˙}{u}}^2+\frac{2u}{\sqrt{u^2+v^2}}\stackrel{˙}{u}\stackrel{˙}{v}+\frac{v}{\sqrt{u^2+v^2}}{\stackrel{˙}{v}}^2-\frac{v}{\sqrt{u^2+v^2}}{\stackrel{˙}{\mathrm{ϕ}}}^2 \\ {a}_z=\stackrel{¨}{\mathrm{ϕ}}uv+2v\stackrel{˙}{u}\stackrel{˙}{\mathrm{ϕ}}+2v\stackrel{˙}{v}\stackrel{˙}{\mathrm{ϕ}} \end{gathered}$$

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