jayzhao
- 2
- 2
- Homework Statement
- Given the Cartesian coordinates for a helicoid:
$$x=\rho cos\phi$$
$$y = \rho sin\phi$$
$$z=h\phi/2\pi$$
where ##\rho\in [0,\infty)##, ##\phi\in (-\infty,\infty)##, and h>0.
Set up the variational principle to search for geodesics on the helicoid. Use ##z## as the independent variable and ##\rho(z)## as the unknown function. Find the Euler equation for the ##\rho(z)## and rewrite it in the form
$$\frac{d^{2}\rho}{dz^{2}}=F(\rho,\rho')$$
- Relevant Equations
- Euler equations:
$$\frac{\partial f}{\partial y}-\frac{d}{dx}\left(\frac{\partial f}{\partial y'}\right)$$
For ##f(x,y(x),y'(x))##
When ##f(y(x),y'(x))## only:
$$f-\frac{\partial f}{\partial y'}y'=constant$$
I've got that length of a curve on the surface is:
$$L=\int_{-\infty}^{\infty}\sqrt{1+\frac{4\pi^{2}}{h^{2}}\rho^{2}+\left(\frac{d\rho}{dz}\right)^{2}}dx$$
So the function to extremise is:
$$f(\rho,\rho')=\sqrt{1+\frac{4\pi^{2}}{h^{2}}\rho^{2}+\left(\frac{d\rho}{dz}\right)^{2}}$$
Where ##\rho'=d\rho /dz##
But I don't know how to get this in the form
$$\frac{d^{2}\rho}{dz^{2}}=F(\rho,\rho')$$
since there doesn't seem to be a second derivative in the function anywhere?
$$L=\int_{-\infty}^{\infty}\sqrt{1+\frac{4\pi^{2}}{h^{2}}\rho^{2}+\left(\frac{d\rho}{dz}\right)^{2}}dx$$
So the function to extremise is:
$$f(\rho,\rho')=\sqrt{1+\frac{4\pi^{2}}{h^{2}}\rho^{2}+\left(\frac{d\rho}{dz}\right)^{2}}$$
Where ##\rho'=d\rho /dz##
But I don't know how to get this in the form
$$\frac{d^{2}\rho}{dz^{2}}=F(\rho,\rho')$$
since there doesn't seem to be a second derivative in the function anywhere?