Proof the shortest path on a sphere is the great circle.

[trytodoit]

There are plenty of proofs for the statement, but I do not find one which is not rely on other assumptions. Here are some common proofs of this statement:

https://en.m.wikipedia.org/wiki/Great_Circle#Derivation_of_shortest_paths
This proof require the path to be differentiable, which is not a part of the statement.

https://math.stackexchange.com/questions/1180923/shortest-path-on-a-sphere
There are several proofs in the page, they are rely on the uniqueness of the shortest path. They thought the uniqueness is intuitive, but I cannot persuade myself on this assumption.

Can anyone provide me a strict proof without other assumption of the statement?

 

[HallsofIvy]

The shortest path is NOT unique, if the two points are polar opposites.

 

[trytodoit]

The shortest path is NOT unique, if the two points are polar opposites.

Yes, therefore the proofs assuming uniqueness of the shortest path are not that intuitive as it sounds. Any better proof for this simple statement.

 

[micromass]

I guess the first thing to do is to actually define what you mean with "length". The following link may help http://www.math.psu.edu/petrunin/papers/alexandrov/bbi.pdf

In particular, you might be interested in section 5.2.2 which proves that every shortest path is differentiable.

 

[trytodoit]

Woo, it looks quite hard to proof this statement.

 

[CWatters]

In 2D.. Say you have two points A and B joined by a straight line. If you add another point C to form a triangle then it should be easy to show that the path ACB is longer than AB. Is it the same for a spherical triangle?

 

[Svein]

This is Physics forum! We don't need no stinkin' mathematics!

Select two points on a sphere. Take a string, anchor it on one of the points, ensure that it crosses the other point, and tighten it. When you cannot tighten it more, you have found the shortest way between the points.

 

[micromass]

This is Physics forum! We don't need no stinkin' mathematics!

Select two points on a sphere. Take a string, anchor it on one of the points, ensure that it crosses the other point, and tighten it. When you cannot tighten it more, you have found the shortest way between the points.

That only gives a geodesic though, not generally a shortest path Indeed, you can imagine two points on the sphere close together, and the string going all around the poles.

 

[Svein]

That only gives a geodesic though, not generally a shortest path Indeed, you can imagine two points on the sphere close together, and the string going all around the poles.

Yes, but that is a very unstable situation. If the sphere is of the "no-friction" type that occurs in a typical physics problem, the tiniest shake of your hands will make the string slip around the sphere.

And "The term "geodesic" comes from geodesy, the science of measuring the size and shape of Earth; in the original sense, a geodesic was the shortest route between two points on the Earth's surface " (http://en.wikipedia.org/wiki/Geodesic).

 

[trytodoit]

Yes, but that is a very unstable situation. If the sphere is of the "no-friction" type that occurs in a typical physics problem, the tiniest shake of your hands will make the string slip around the sphere.

And "The term "geodesic" comes from geodesy, the science of measuring the size and shape of Earth; in the original sense, a geodesic was the shortest route between two points on the Earth's surface " (http://en.wikipedia.org/wiki/Geodesic).

In my opinion, you cannot define the shortest path by tight the string. For the sphere, it happens to be only two stable states, as mentioned by @micromass, but generally you still cannot say a stable state of the tight string is the shortest path for all kinds of shapes. Even worse, according to the pdf provided by @micromass, for general shapes, the shortest path maybe non-smooth and not unique. Tight the string may be useful in finding a local extremes, but not quite helpful for finding the shortest path.

 

[Svein]

The shortest path is NOT unique, if the two points are polar opposites.

The title of this thread is "proof that the shortest path on a sphere is the great circle". It does not say anything about uniqueness. On the other hand, the title is incorrect as it stands. A better title would be: "Proof that the shortest path between two points on a sphere is part of a great circle".