What are some tips for finding geodesics on a surface in R^3?

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Discussion Overview

The discussion revolves around finding geodesics on surfaces in R^3, specifically focusing on the sphere and cylinder. Participants explore methods for demonstrating that certain curves, like great circles, are geodesics, and share their experiences with the challenges involved in the calculations and conceptual understanding.

Discussion Character

  • Exploratory
  • Technical explanation
  • Conceptual clarification
  • Debate/contested
  • Homework-related

Main Points Raised

  • One participant suggests that great circles on the sphere are intuitively geodesics but finds the mathematical proof involving Christoffel symbols and differential equations challenging.
  • Another participant proposes that it might be easier to show that great circles satisfy the geodesic equation directly.
  • The original poster acknowledges the suggestion and expresses a need to make geometric arguments to demonstrate that great circles are the only geodesics.
  • The original poster later expresses confusion and requests more explicit tips for calculating geodesics, indicating difficulty with the symbology involved.
  • Ultimately, the original poster shares a personal finding that for a 2-dimensional surface in R^3, all geodesics have second derivatives normal to the surface, noting that this approach does not extend to higher dimensions.

Areas of Agreement / Disagreement

Participants do not reach a consensus on the best method for demonstrating geodesics, and multiple approaches are discussed without resolution. The original poster's findings are presented as personal insights rather than universally accepted conclusions.

Contextual Notes

The discussion reflects limitations in understanding the mathematical framework and the specific challenges faced when applying concepts to different surfaces. There is also an acknowledgment that the findings may not apply in higher dimensions.

Feles Cestriana
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The Great Circle et al...

It is certainly intuitively clear that the geodesics on the sphere are the great circles. But, showing that seems to be a bit tricky...

I tried working out the Christoffel symbols, and then plugging that into the differential equations for the geodesics, and aside from it being a bit of a mess, I'm up against a bit of a wall with the system.

Now, perhaps I just need to grind it a little more, and it will all fall out, but, I can't help but wonder if there is a better way to "show" what the geodesics of a surface are.

I've had the same issue with the cylinder...

Any tips, to just point me in the right direction, I'd be most appreciative.

Thanks,

-FC
 
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Wouldn't it be easier to just show that great circles satisfy the geodesic equation?

- Warren
 
Hmmm. Yes, I suppose it might be. Thank you. I will try that.

But I think I will have to make some geometric arguments to show that those are all the geodesics.

Thanks for the tip. Is this the typical method of attack for these sorts of problems?

Thanks again.

-FC
 
Hmmm... okay, well I'm still having a bit of trouble. Are there any slightly more explicit tips to be had? I'm a bit lost in the symbology, and seem to be unable to calculate an answer here.

Thanks,

-FC
 
oooookay...

Well, since no one has given any further assistance, I had to figure this out for myself.

And just in case someone else out there reads this thread in hopes of finding something out, I'll post my findings, so this was not all in vain.

For a 2-dimensional surface in R^3, all geodesics will have second derivatives normal to the surface.

This is the easiest way to deal with geodesics on a surface in R^3.

Note that this doesn't work in higher dimentions.

I hope you all have better luck than I have had trying to figure this stuff out.

-FC
 

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