Deriving the Geodesic formulae

In summary, the geodesic formula is a mathematical equation that calculates the shortest path between two points on a curved surface, taking into account the surface's curvature. It is derived using principles of differential geometry and has various applications in fields such as geodesy, navigation, and cartography. Some challenges in deriving the formula include accurately representing surface curvature and dealing with irregular surfaces. Alternative methods, such as numerical approximation techniques, exist for calculating geodesic curves.
  • #1
PhysicistFTW
2
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1. A straight line in flat space may be defined by the equation:
(when I use the ^ symbol in this case it means like upper subscript not to the power)


(U^v)(d/dx^v)(U^u)=0


(U^u=dx^u/ds)

derive the geodesic equation.

Please help I'm completely clueless all I can really see to do is mix the two equations but that doesn't really show me anything more clearly, any help would be greatly appreciated.


 
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  • #2
also it says to make the equation covariant if that helps
 

1. What is the geodesic formula?

The geodesic formula is a mathematical equation that calculates the shortest path between two points on a curved surface, such as the Earth or a sphere. It takes into account the curvature of the surface and provides a geodesic curve that minimizes the distance between the two points.

2. How is the geodesic formula derived?

The geodesic formula is derived using the principles of differential geometry, specifically the concept of a geodesic curve. This curve is defined as the path of least resistance on a curved surface, and the formula is derived by solving for the tangent vector of this curve at each point.

3. What are the applications of the geodesic formula?

The geodesic formula has many applications in various fields such as geodesy, navigation, and cartography. It is used to calculate the most efficient routes for transportation, to determine the shortest path between two points on a curved surface, and to create accurate maps and globes.

4. What are some challenges in deriving the geodesic formula?

One of the main challenges in deriving the geodesic formula is accurately representing the curvature of a surface. This requires a deep understanding of differential geometry and the ability to apply complex mathematical concepts. Additionally, real-world surfaces are often irregular and require advanced techniques to accurately calculate the geodesic curve.

5. Are there any alternative methods for calculating geodesic curves?

Yes, there are alternative methods for calculating geodesic curves, such as numerical approximation techniques. These methods involve dividing the curved surface into smaller segments and using mathematical algorithms to approximate the geodesic curve. While they may not be as precise as the geodesic formula, they can be useful for practical applications.

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