91Ó°ÊÓ

The motion of as particle acted on by force is$F=∇V.$.

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

Formula for the Lagrange of the system in cylindrical coordinates is mentioned below.

$$\begin{gathered} L=\frac{1}{2}{\stackrel{Ë™}{mr}}^2-V\left(r,\mathrm{Ï•},z\right) \\ L=\stackrel{¨}{r{\stackrel{âˆ\S }{e}}_r}+\stackrel{Ë™}{r}\mathrm{Ï•}\mathrm{θ}-\stackrel{Ë™}{r}{\mathrm{Ï•}}^2{\stackrel{âˆ\S }{e}}_{\mathrm{Ï•}}+\stackrel{Ë™}{r}\mathrm{Ï•}{\stackrel{âˆ\S }{e}}_z \\ L=\frac{1}{2}m\left(r^2+r^2{\mathrm{Ï•}}^2+{\stackrel{Ë™}{z}}^2\right)-V\left(r,\mathrm{Ï•},z\right) \end{gathered}$$

Apply Euler Lagrange’s on the equation mentioned above

$$\begin{gathered} \frac{d}{dt}\left(\frac{∂L}{∂r}\right)=\stackrel{¨}{m}r=m\stackrel{˙}{r}{\mathrm{θ}}^2+mr{\sin }^2\stackrel{˙}{\mathrm{θ}}{\mathrm{ϕ}}^2-\frac{∂V}{∂r} \\ \frac{d}{dt}\left(\frac{∂L}{∂r}\right)=m{r}^{\stackrel{¨}{2}}\mathrm{θ}=2m\stackrel{˙}{r}\stackrel{˙}{\mathrm{θ}}=m{r}^2\sin \stackrel{˙}{\mathrm{θ}}{\mathrm{ϕ}}^2-\frac{∂V}{∂r} \\ \frac{d}{dt}\left(\frac{∂L}{∂\mathrm{ϕ}}\right)=mr\frac{d}{dt}\left(r^2{\sin }^2\mathrm{θ}\mathrm{ϕ}\right)=-\frac{∂V}{∂\mathrm{ϕ}} \end{gathered}$$

Divide the equation with respective scale factors given below.

$$\begin{gathered} h_{\mathrm{Ï•}}=r \\ {h}_r=1 \\ {h}_z=1 \end{gathered}$$

The equation becomes as follows.

$$\begin{gathered} m\left(r-\stackrel{˙}{r}{\mathrm{θ}}^2-r{\sin }^2\mathrm{θ}{\mathrm{ϕ}}^2\right)=-\frac{∂V}{∂r} \\ m\left(\stackrel{¨}{r}\mathrm{θ}+2\stackrel{˙}{r}\mathrm{θ}-r\sin \mathrm{θ}{\mathrm{ϕ}}^2\right)=-\frac{1}{r}\frac{∂V}{∂\mathrm{θ}} \\ m\left(r\sin \stackrel{¨}{\mathrm{θ}\mathrm{ϕ}}+2\sin \mathrm{θ}\stackrel{˙}{r}\mathrm{ϕ}+2r\cos \stackrel{˙}{\mathrm{θ}}\stackrel{˙}{\mathrm{θ}}\mathrm{ϕ}\right)=-\frac{1}{r\sin \mathrm{θ}}\frac{∂V}{∂\mathrm{ϕ}} \end{gathered}$$

The components of acceleration are mentioned below.

$$\begin{gathered} a_r=\stackrel{¨}{r}-\stackrel{˙}{r}{\mathrm{θ}}^2-\stackrel{˙}{r}{\mathrm{ϕ}}^2\mathrm{θ} \\ {a}_{\mathrm{θ}}=\stackrel{¨}{r}\mathrm{θ}+\stackrel{˙}{2}\stackrel{˙}{\mathrm{θ}}\mathrm{θ}-\stackrel{˙}{r}{\mathrm{ϕ}}^2\sin \mathrm{θ}\cos \mathrm{θ} \\ {a}_{\mathrm{ϕ}}=r\sin \stackrel{¨}{\mathrm{θ}}\mathrm{ϕ}+2\sin \stackrel{˙}{\mathrm{θ}\stackrel{˙}{r}\mathrm{ϕ}}+2r\cos \stackrel{˙}{\mathrm{θ}}\stackrel{˙}{\mathrm{θ}}\mathrm{ϕ} \end{gathered}$$