Timeline for Geodesics on spheres are great circles
Current License: CC BY-SA 2.5
6 events
| when toggle format | what | by | license | comment | |
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| Jan 19, 2010 at 6:42 | comment | added | Ryan Budney | I like to get my differential geometry students to prove great circles are length-minimizing using the Cauchy-Crofton theorem. That's kind of over-the-top but it's an entertaining argument. It also works in Euclidean and hyperbolic geometry, like the symmetry argument of Mariano's or, oh, that's the same as my comment to Jose's reply, too. | |
| Jan 19, 2010 at 6:04 | comment | added | Anirbit | Its very gratifying to get back such detailed expository answers! Thanks a lot. I was doing the Euler-Lagrange equations on the function $L = \dot{\theta}^2 + sin^2(\theta)\dot{\phi}^2$. Isn't that the same as doing it on the energy functional as you have suggested since the Energy Functional as you state is integral of my $L$. Your suggestion of calculation 1 is something I had done earlier. But how does that help in proving great circles are geodesics? Is there any canonical way of parameterizing the great circles as unit speed curves? (same was the spirit of Jose's answer) | |
| Jan 19, 2010 at 5:50 | vote | accept | Anirbit | ||
| Jan 18, 2010 at 19:46 | comment | added | Deane Yang | Agreed! I tend to shortchange direct geometric arguments, but they are definitely worth learning, too. | |
| Jan 18, 2010 at 18:52 | comment | added | Ryan Budney | There are direct arguments as well -- many textbooks have standardized arguments that the shortest curve in Euclidean space connecting two points is a straight line. The primary tool is the triangle inequality. You could do the same on the sphere, using the sphere's intrinsic metric. Alternatively, there are cute proofs using the Cauchy-Crofton theorem for spherical, euclidean and hyperbolic geometry. | |
| Jan 18, 2010 at 18:45 | history | answered | Deane Yang | CC BY-SA 2.5 |