91Ó°ÊÓ

A Bipolar.

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.

Velocity in bipolar is $\stackrel{Ë™}{s}=\stackrel{Ë™}{u}\frac{a}{\cos hu+\cos v}{\hat{e}}_u+\stackrel{Ë™}{\stackrel{Ë™}{v}\frac{a}{\cos hu+\cos v}{\hat{e}}_u}$.

Find the acceleration.

$$\begin{gathered} \stackrel{¨}{s}=(\stackrel{¨}{u}+{\mathrm{Γ}}_{\mathrm{ij}}^{\mathrm{u}}\stackrel{˙}{{\mathrm{q}}_{\mathrm{i}}{\stackrel{˙}{\mathrm{q}}}_{\mathrm{j}}){\mathrm{h}}_{\mathrm{u}}{\hat{\mathrm{e}}}_{\mathrm{u}}+}(\stackrel{¨}{\mathrm{u}}+{\mathrm{Γ}}_{\mathrm{ij}}^{\mathrm{v}}\stackrel{˙}{{\mathrm{q}}_{\mathrm{i}}{\stackrel{˙}{\mathrm{q}}}_{\mathrm{j}}){\mathrm{h}}_{\mathrm{v}}{\hat{\mathrm{e}}}_{\mathrm{v}}} \\ \frac{{∂}^2\mathrm{s}}{∂{\mathrm{q}}_{\mathrm{i}}∂{\mathrm{q}}_{\mathrm{j}}}={\mathrm{Γ}}_{\mathrm{ij}}^{\mathrm{k}}\frac{∂\mathrm{s}}{∂{\mathrm{q}}_{\mathrm{k}}} \end{gathered}$$

Solve further.

$$\begin{gathered} \frac{{∂}^{2}s}{∂u^2}=-\frac{a\sin hu}{\cos hu+\cos v}\frac{{\sin }^{2}v+1+\cos hu+\cos v}{{\left(\cos hu+\cos v\right)}^2}{\hat{e}}_x+\frac{a\sin v}{\cos hu+\cos v}\frac{{\sin }^{2}u-(1+\cos u+\cos v)}{{\left(\cos hu+\cos v\right)}^2}{\hat{e}}_y \\ \frac{{∂}^{2}s}{∂u^2}=-\frac{a\sin hu}{\cos u+\cos v}\frac{∂s}{∂u}-\frac{\sin v}{\cos hu+\cos v}\frac{∂s}{∂u} \\ \frac{{∂}^{2}s}{∂u∂v}=-\frac{a\sin v}{\cos hu+\cos v}\frac{{\sin }^{2}u+(1+\cos hu+\cos v)}{{\left(\cos hu+\cos v\right)}^2}{\hat{e}}_x+\frac{a\sin hu}{\cos hu+\cos v}\frac{{\sin }^{2}v-1+\cos u+\cos v}{{\left(\cos hu+\cos v\right)}^2}{\hat{e}}_y \\ \frac{{∂}^{2}s}{∂u∂v}=\frac{\sin v}{\cos hu+\cos v}\frac{∂s}{∂u}-\frac{\sin hu}{\cos hu+\cos v}\frac{∂s}{∂u} \end{gathered}$$

Solve further.

$$\begin{gathered} \frac{{∂}^{2}s}{∂v^2}=\frac{a\sin hu}{\cos hu+\cos v}\frac{{\sin }^{2}v+1+\cos hu+\cos v}{{\left(\cos hu+\cos v\right)}^2}{\hat{e}}_x+\frac{a\sin v}{\cos hu+\cos v}\frac{1+\cos hu\cos v-\sin h^{2}u}{{\left(\cos hu+\cos v\right)}^2}\hat{e} \\ \frac{{∂}^{2}s}{∂v^2}=\frac{a\sin hu}{\cos hu+\cos v}\frac{∂s}{∂u}+\frac{\sin v}{\cos hu+\cos v}\frac{∂s}{∂u} \end{gathered}$$

Find the components of acceleration.

$$\begin{gathered} {\tilde{S}}_u=\left(\stackrel{¨}{u}-\frac{\sin hu}{\cos hu+\cos v}{\stackrel{˙}{u}}^2+\frac{2\sin v}{\cos hu+\cos v}\stackrel{˙}{u}\stackrel{˙}{v}+\frac{\sin hu}{\cos hu+\cos v}{\stackrel{˙}{v}}^2\right)\frac{a}{\cos hu+\cos v} \\ {\tilde{S}}_v=\left(\stackrel{¨}{u}-\frac{\sin v}{\cos hu+\cos v}{\stackrel{˙}{u}}^2+\frac{2\sin hu}{\cos hu+\cos v}\stackrel{˙}{u}\stackrel{˙}{v}+\frac{\sin v}{\cos hu+\cos v}{\stackrel{˙}{v}}^2\right)\frac{a}{\cos hu+\cos v} \end{gathered}$$