Help with the Euler-Lagrange formula for a geodesic

In summary, the Euler-Lagrange formula is a mathematical equation used to calculate the shortest distance between two points on a curved surface. It is derived from calculus of variations and can be applied to any curved surface as long as the metric is defined. The formula has significant applications in various fields and helps us understand principles of curvature and shortest distance. However, it may not accurately represent surfaces with sharp edges or discontinuities and can become more complex for complex surfaces.
  • #1
martinhiggs
24
0

Homework Statement



The metric is:

ds[tex]^{2}[/tex] = y[tex]^{2}[/tex](dx[tex]^{2}[/tex] + dy[tex]^{2}[/tex])

I have to find the equation relating x and y along a geodesic.

The Attempt at a Solution



ds = [tex]\sqrt{ydx^{2} + ydy^{2}}[/tex] - is this right?

ds = [tex]\sqrt{y + yy'^{2}}[/tex] dx

F = [tex]\sqrt{y + yy'^{2}}[/tex]

So then I apply the Euler-Lagrange equation

dF/dy - d/dx[dF/dy'] = 0Now I'm stuck, please help.
 
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  • #2
It's a y^2 inside the square root, isn't it? Not a y.
 

1. What is the Euler-Lagrange formula for a geodesic?

The Euler-Lagrange formula for a geodesic is a mathematical equation that describes the path of shortest distance between two points on a curved surface, such as a sphere or a curved plane. It is derived from the principles of calculus of variations and is used in fields such as physics and differential geometry.

2. How do you use the Euler-Lagrange formula to find a geodesic?

To find a geodesic using the Euler-Lagrange formula, you first need to define the metric of the curved surface. This involves specifying the distance between two points on the surface. Then, using the formula, you can calculate the shortest path between two points, also known as the geodesic.

3. Can the Euler-Lagrange formula be applied to any curved surface?

Yes, the Euler-Lagrange formula can be applied to any curved surface as long as the metric of the surface is defined. This includes surfaces such as spheres, cylinders, and even more complex curved surfaces.

4. What is the significance of the Euler-Lagrange formula in science?

The Euler-Lagrange formula is significant in science because it allows us to calculate the shortest distance between two points on a curved surface, which has numerous applications in fields such as physics, engineering, and computer graphics. It also helps us understand the principles of curvature and shortest distance in mathematics and geometry.

5. Are there any limitations to using the Euler-Lagrange formula for finding geodesics?

One limitation of using the Euler-Lagrange formula for geodesics is that it assumes the surface is smooth and continuous. This means that it may not accurately represent the path of shortest distance on surfaces with sharp edges or discontinuities. Additionally, the formula can become more complex and challenging to solve for more complex surfaces.

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