What is the function to extremise for finding geodesics on a Helicoid?

In summary, the conversation discusses the extremization of a function involving a curve on a surface using Euler equations and the first integral equation. The speaker is unsure of how to get the function in the form of a second derivative and the other person suggests using the Euler equation instead. They also realize that the first integral equation and the Euler equation are two different things.
  • #1
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?
 
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  • #2
jayzhao said:
Relevant Equations:: Euler equations:
$$\frac{\partial f}{\partial y}-\frac{d}{dx}\left(\frac{\partial f}{\partial y'}\right)$$When ##f(y(x),y'(x))## only:$$f-\frac{\partial f}{\partial y'}y'=constant$$

Maybe they want you to use the Euler equation ##\frac{\partial f}{\partial y}-\frac{d}{dx}\left(\frac{\partial f}{\partial y'}\right)## rather than the "first-integral" equation ##f-\frac{\partial f}{\partial y'}y'= \rm {const}##.
 
Last edited:
  • #3
TSny said:
Maybe they want you to use the Euler equation ##\frac{\partial f}{\partial y}-\frac{d}{dx}\left(\frac{\partial f}{\partial y'}\right)## rather than the "first-integral" equation ##f-\frac{\partial f}{\partial y'}y'= \rm {const}##.
Thank you! I think you're right that's what they wanted. I was getting confused because I thought the first integral equation was a "special case" of the Euler equation but it turns out they're two different things.
 
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  • #4
You can always differentiate the Beltrami identity to recover the Euler-Lagrange form (as long as your problem is one-dimensional).
 
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Related to What is the function to extremise for finding geodesics on a Helicoid?

1. What is the definition of a geodesic on a Helicoid?

A geodesic on a Helicoid is a curve that follows the shortest path between two points on the surface of the Helicoid. It can also be described as a curve that is locally straight and has zero geodesic curvature.

2. What is the function used to extremise for finding geodesics on a Helicoid?

The function used to extremise for finding geodesics on a Helicoid is the arc length functional, which is given by the integral of the square root of the first fundamental form of the Helicoid.

3. How is the arc length functional used to find geodesics on a Helicoid?

The arc length functional is used to find geodesics on a Helicoid by minimizing the functional with respect to the path of the geodesic. This is done by setting the variation of the functional equal to zero and solving the corresponding Euler-Lagrange equations.

4. What are the Euler-Lagrange equations for finding geodesics on a Helicoid?

The Euler-Lagrange equations for finding geodesics on a Helicoid are a set of second-order differential equations that describe the path of the geodesic. They can be derived by setting the variation of the arc length functional equal to zero and solving for the corresponding equations.

5. Are there any other methods for finding geodesics on a Helicoid?

Yes, there are other methods for finding geodesics on a Helicoid, such as using the geodesic equation or the Jacobi equation. These methods involve solving differential equations and can be used to find geodesics on more complicated surfaces as well.

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