Parabolic.

A Parabolic.

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 parabolic is $\dot{\mathrm{s}}=\dot{\mathrm{u}}\sqrt{{\mathrm{u}}^2+{\mathrm{v}}^2}{\hat{\mathrm{e}}}_{\mathrm{u}}+\dot{\mathrm{v}}\sqrt{{\mathrm{u}}^2+{\mathrm{v}}^2}{\hat{\mathrm{e}}}_{\mathrm{v}}+\dot{\mathrm{\varphi uv}}{\hat{\mathrm{e}}}_{\mathrm{\varphi }}$.

Find the acceleration.

$$\begin{gathered} \ddot{\mathrm{s}}=\left(\ddot{\mathrm{u}}+{\mathrm{\Gamma }}_{\mathrm{ij}}^{\mathrm{u}}{\dot{\mathrm{q}}}_{\mathrm{i}}{\dot{\mathrm{q}}}_{\mathrm{j}}\right){\mathrm{h}}_{\mathrm{u}}{\hat{\mathrm{e}}}_{\mathrm{u}}+\left(\ddot{\mathrm{v}}+{\mathrm{\Gamma }}_{\mathrm{ij}}^{\mathrm{v}}{\dot{\mathrm{q}}}_{\mathrm{i}}{\dot{\mathrm{q}}}_{\mathrm{j}}\right){\mathrm{h}}_{\mathrm{u}}{\hat{\mathrm{e}}}_{\mathrm{u}}+\left(\ddot{\mathrm{\varphi }}+{\mathrm{\Gamma }}_{\mathrm{ij}}^{\mathrm{\varphi }}{\dot{\mathrm{q}}}_{\mathrm{i}}{\dot{\mathrm{q}}}_{\mathrm{j}}\right){\mathrm{h}}_{\mathrm{\varphi }}{\hat{\mathrm{e}}}_{\mathrm{\varphi }}. \\ \frac{{\partial }^2\mathrm{s}}{\partial {\mathrm{q}}_{\mathrm{i}}\partial {\mathrm{q}}_{\mathrm{j}}}={\mathrm{\Gamma }}_{\mathrm{ij}}^{\mathrm{k}}\frac{\partial \mathrm{s}}{\partial {\mathrm{q}}_{\mathrm{k}}} \end{gathered}$$

Solve further.

$$\begin{gathered} \frac{{\partial }^{2}s}{\partial u^2}={\hat{e}}_z\frac{u}{u^2+v^2}\frac{\partial s}{\partial u}-\frac{v}{u^2+v^2}\frac{\partial s}{\partial v} \\ \frac{{\partial }^{2}s}{\partial u\partial v}=\frac{v}{u^2+v^2}\frac{\partial s}{\partial u}+\frac{u}{u^2+v^2}\frac{\partial s}{\partial v} \\ \frac{{\partial }^{2}s}{\partial v^2}=-\frac{v}{u^2+v^2}\frac{\partial s}{\partial u}+\frac{u}{u^2+v^2}\frac{\partial s}{\partial v} \\ \frac{{\partial }^{2}s}{\partial u\partial \varphi }=\frac{1}{s}\frac{\partial s}{\partial \varphi } \end{gathered}$$

Solve further.

role="math" localid="1659335844957" $$\begin{gathered} \frac{{\partial }^{2}s}{\partial v\partial \varphi }=\frac{1}{v}\frac{\partial s}{\partial \varphi } \\ \frac{{\partial }^{2}s}{\partial {\varphi }^2}=-\frac{u{v}^2}{v^2+u^2}\frac{\partial s}{\partial u}-\frac{u^{2}v}{v^2+u^2}\frac{\partial s}{\partial v} \end{gathered}$$

Find the components of acceleration.

$$\begin{gathered} {\ddot{s}}_u=\left(\ddot{u}+\frac{u}{u^2+v^2}{\dot{u}}^2+\frac{2v}{u^2+v^2}\dot{u}\dot{v}+\frac{u}{u^2+v^2}{\dot{v}}^2-\frac{u{v}^2}{u^2+v^2}{\dot{\varphi }}^2\right)\sqrt{u^2+v^2} \\ {\ddot{s}}_v=\left(\ddot{u}+\frac{u}{u^2+v^2}{\dot{u}}^2+\frac{2u}{u^2+v^2}\dot{u}\dot{v}+\frac{v}{u^2+v^2}{\dot{v}}^2-\frac{u^{2}v}{u^2+v^2}{\dot{\varphi }}^2\right)\sqrt{u^2+v^2}.6 \\ {\ddot{s}}_{\varphi }=\left(\ddot{\varphi }+\frac{2}{u}\dot{u}\dot{\varphi }+\frac{2}{u}\dot{v}\dot{\varphi }\right)uv \end{gathered}$$