91Ó°ÊÓ

Equation of cone is given as $x^2+y^2=z^2$. Geodesics on the cone is to be found out using cylindrical coordinates.

In the calculus of variations andclassical mechanics, the Euler equations is a system of second-orderordinary differential equations whose solutions arestationary points of the givenaction functional.

Geodesics on a cone is to be found out using cylinderical coordinates.

Equation of cone is

$$\begin{gathered} x^2+y^2=z^2 \\ {r}^2=z^2 \\ r=z \\ dr=dz \end{gathered}$$

So distance integral is to be minimized.

$$\begin{gathered} ∫ds=∫\sqrt{d{r}^2+d{z}^2+r^{2}d{\mathrm{θ}}^2} \\ =∫\sqrt{2d{r}^2+r^{2}d\mathrm{θ}} \\ =∫\sqrt{2+r^2\mathrm{θ}{\text{'}}^2}dr \\ \mathrm{θ}\text{'}=\frac{d\mathrm{θ}}{dr} \end{gathered}$$

Let $F=\sqrt{2+r^2\mathrm{θ}{\text{'}}^2}$

Euler equation for coordinateslocalid="1664890346730" $\left(r,\mathrm{θ}\right)$islocalid="1664891513287" $\frac{d}{dr}\left(\frac{∂F}{∂\mathrm{θ}\text{'}}\right)-\frac{∂F}{∂\mathrm{θ}}=0$

Calculate the required derivatives

$$\begin{gathered} \frac{∂F}{∂\mathrm{θ}\text{'}}=\frac{r^2\mathrm{θ}\text{'}}{\sqrt{2+r^2\mathrm{θ}{\text{'}}^2}} \\ \frac{∂F}{∂\mathrm{θ}}=0 \end{gathered}$$\

Therefore,

$$\begin{gathered} \frac{d}{dr}\left[\frac{r^2\mathrm{θ}\text{'}}{\sqrt{2+r^2\mathrm{θ}{\text{'}}^2}}\right]=0 \\ \frac{r^2\mathrm{θ}\text{'}}{\sqrt{2+r^2\mathrm{θ}{\text{'}}^2}}=C \\ \mathrm{θ}{\text{'}}^2=\frac{2C^2}{r^2\sqrt{\left(r^2-C^2\right)}} \\ \mathrm{θ}\text{'}=\frac{\sqrt{2}C}{r\sqrt{r^2-C^2}} \end{gathered}$$

Where$C$is constant.

Integrate $\mathrm{θ}\text{'}=\frac{\sqrt{2}C}{r\sqrt{r^2-C^2}}$to get the desired result

$$\begin{gathered} \mathrm{θ}\text{'}=∫\frac{\sqrt{2}C}{r\sqrt{r^2-C^2}}dr \\ =\sqrt{2}arc\tan \left(\frac{\sqrt{r^2-C^2}}{C}\right)+B \end{gathered}$$

Therefore,$\mathrm{θ}=\sqrt{2}arc\tan \left(\frac{\sqrt{r^2-C^2}}{C}\right)+B$, where C is constant and B is the integration constant.