Question

Elliptical cylinder.

Short answer

The required values are mentioned below.

$$\begin{gathered} {\ddot{s}}_u=\left(\ddot{u}+\frac{\sin hu\cos hu}{{\sin }^{2}u+{\sin }^{2}v}{\dot{u}}^2+\frac{2\sin u\cos u}{{\sin }^{2}u+{\sin }^{2}v}\dot{u}\dot{v}-\frac{\sin u\cos u}{{\sin }^{2}u+{\sin }^{2}v}{\dot{v}}^2\right)h_u \\ {\ddot{s}}_v=\left(\ddot{u}+\frac{\sin v\cos u}{{\sin }^{2}u+{\sin }^{2}v}{\dot{u}}^2+\frac{2\sin hv\cos hu}{{\sin }^{2}u+{\sin }^{2}v}\dot{u}\dot{v}-\frac{\sin v\cos u}{{\sin }^{2}u+{\sin }^{2}v}{\dot{v}}^2\right)h_u \\ {\ddot{s}}_z=\ddot{z} \end{gathered}$$

Step-by-step solution

Given Information

An elliptical cylinder.

Definition of cylindrical coordinates.

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.

Find all the values.

Velocity in elliptical cylinder is$\dot{s}=\dot{u}a\sqrt{\sin h^{2}u+{\sin }^{2}v}{\hat{e}}_u+\dot{v}a\sqrt{\sin h^{2}u+{\sin }^{2}v}{\hat{e}}_v+\dot{z{\hat{e}}_z}$.

Find the acceleration.

localid="1659333210449" $$\begin{gathered} \ddot{s}=\left(\ddot{u}+{\Gamma }_{ij}^u{\dot{q}}_i{\dot{q}}_j\right)h_u{\hat{e}}_u+\left(\ddot{v}+{\Gamma }_{ij}^v{\dot{q}}_i{\dot{q}}_j\right)h_v{\hat{e}}_u+\ddot{z}{\hat{e}}_z \\ \frac{{\partial }^{2}s}{\partial q_i\partial q_j}={\Gamma }_{ij}^k\frac{\partial s}{\partial q_k} \end{gathered}$$

Solve further.

$$\begin{gathered} \frac{{\partial }^2\mathrm{s}}{\partial {\mathrm{u}}^2}=\mathrm{acoshucosv}{\hat{\mathrm{e}}}_{\mathrm{x}}+\mathrm{asinhusinv}{\hat{\mathrm{e}}}_{\mathrm{y}} \\ \frac{{\partial }^2\mathrm{s}}{\partial \mathrm{u}\partial \mathrm{v}}=-\mathrm{asinhusinv}{\hat{\mathrm{e}}}_{\mathrm{x}}+\mathrm{acoshucosv}{\hat{\mathrm{e}}}_{\mathrm{y}} \\ \frac{{\partial }^2\mathrm{s}}{\partial {\mathrm{v}}^2}=\mathrm{acoshucosv}{\hat{\mathrm{e}}}_{\mathrm{x}}-\mathrm{asinhusinv}{\hat{\mathrm{e}}}_{\mathrm{y}} \end{gathered}$$

Find the value of ${\hat{e}}_x$and ${\hat{e}}_y$.

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

Find the coefficients.

$$\begin{gathered} \frac{{\partial }^2\mathrm{s}}{\partial {\mathrm{u}}^2}=\frac{\mathrm{sinhucoshu}}{{\sinh }^2\mathrm{u}+{\sin }^2\mathrm{v}}\frac{\partial \mathrm{s}}{\partial \mathrm{u}}-\frac{\mathrm{sinvcosv}}{{\sinh }^2\mathrm{u}+{\sin }^2\mathrm{v}}\frac{\partial \mathrm{s}}{\partial \mathrm{v}} \\ {\mathrm{\Gamma }}_{\mathrm{uu}}^{\mathrm{u}}=\frac{\mathrm{sinhucoshu}}{{\sinh }^2\mathrm{u}+{\sin }^2\mathrm{v}} \\ {\mathrm{\Gamma }}_{\mathrm{uu}}^{\mathrm{u}}=-\frac{\mathrm{sinvcosv}}{{\sinh }^2\mathrm{u}+{\sin }^2\mathrm{v}} \\ \frac{{\partial }^2\mathrm{s}}{\partial {\mathrm{u}}^2}=\frac{\mathrm{sinvcosv}}{{\sinh }^2\mathrm{u}+{\sin }^2\mathrm{v}}\frac{\partial \mathrm{s}}{\partial \mathrm{u}}-\frac{\mathrm{sinhucoshu}}{{\sinh }^2\mathrm{u}+{\sin }^2\mathrm{v}}\frac{\partial \mathrm{s}}{\partial \mathrm{v}} \end{gathered}$$

Find the further coefficients.

$$\begin{gathered} {\Gamma }_{uv}^u=\frac{\sin u\cos u}{\sin h^{2}u+{\sin }^{2}v};{\Gamma }_{uv}^u=\frac{\sin hv\cos hv}{\sin h^{2}u+{\sin }^{2}v} \\ \frac{{\partial }^{2}s}{\partial v^2}=-\frac{\sin hu\cos hu}{\sin h^{2}u+{\sin }^{2}v}\frac{\partial s}{\partial u}+\frac{\sin v\cos v}{\sin h^{2}u+{\sin }^{2}v}\frac{\partial s}{\partial v} \\ {\Gamma }_{vv}^u=-\frac{\sin hu\cos hu}{\sin h^{2}u+{\sin }^{2}v} \\ {\Gamma }_{vv}^v=-\frac{\sin v\cos v}{\sin h^{2}u+{\sin }^{2}v} \end{gathered}$$

Find the components of acceleration.

$$\begin{gathered} {\ddot{s}}_u=\left(\ddot{u}+\frac{\sin hu\cos hu}{{\sin }^{2}u+{\sin }^{2}v}{\dot{u}}^2+\frac{2\sin u\cos u}{{\sin }^{2}u+{\sin }^{2}v}\dot{u}\dot{v}-\frac{\sin u\cos u}{{\sin }^{2}u+{\sin }^{2}v}{\dot{v}}^2\right)h_u \\ {\ddot{s}}_v=\left(\ddot{u}+\frac{\sin v\cos u}{{\sin }^{2}u+{\sin }^{2}v}{\dot{u}}^2+\frac{2\sin hv\cos hu}{{\sin }^{2}u+{\sin }^{2}v}\dot{u}\dot{v}-\frac{\sin v\cos u}{{\sin }^{2}u+{\sin }^{2}v}{\dot{v}}^2\right)h_u \\ {\ddot{s}}_z=\ddot{z} \end{gathered}$$.