Timeline for Is there a coordinate-free proof of the hamiltonian character of the geodesic flow?
Current License: CC BY-SA 3.0
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| Feb 18, 2012 at 7:39 | comment | added | agt | Dear Alvarezpaiva, thanks a lot for your answer. Excuse me if my question was prone to be misunderstood, because I should have claimed as starting point my understanding of geodesic flow. Taking the coordinate expression of the equation for geodesic curve (as Def.1.4.2 in Jost) I recognize them as the projection on the base of the geodesic spray: \emph{the unique vector field on $TM$ which is horizontal and special} (i.e. second order edo on $M$) | |
| Feb 17, 2012 at 6:55 | comment | added | alvarezpaiva | @Paul Skerrit: Thanks for the comment and now I think I misread the problem. It seems to me that Giuseppe's question has two parts to it, each with a well-known answer: (1) a coordinate-free proof of the first variation formula in a pseudo-Riemannian manifold can be found in lots of places. For example in Jost's book "Riemannian Geometry and Geometric Analysis"; (2) A coordinate-free proof that the Euler-Lagrange equations of a regular Lagrangian are Hamiltonian (using Cartans formula for Lie derivatives) is also quite classical, but I'll look for a good reference. | |
| Feb 17, 2012 at 6:11 | comment | added | user17945 | BTW apologies for the bump - accidentally posted an answer instead of a comment - doh! | |
| Feb 17, 2012 at 6:05 | comment | added | user17945 | Viewing things this way, one still needs to demonstrate that the Euler-Lagrange equations are the geodesic equations for the metric. Abraham and Marsden demonstrate this in Theorem 3.7.1 (page 224), but using coordinates, which Giuseppe was trying to avoid. It seems that showing this invariantly would still require the introduction of a connection, no? (Presumably the connection mentioned by A&M in the remarks on the bottom of page 227 is the same one introduced by the two posters above - though I would need to check to be sure.) | |
| Feb 17, 2012 at 4:43 | comment | added | Theo Johnson-Freyd | With @Stefan, I highly appreciate the Marsden posts his books online, and I hope others do too. But I also think @alvarezpaiva should elaborate the answer, for two reasons: (1) it is good practice to write such elaborations, (2) the few mouse-clicks, the better. | |
| Feb 16, 2012 at 20:17 | comment | added | Stefan Waldmann | @Deane: you might like this (even legal) way to take a look cds.caltech.edu/~marsden/books/Foundations_of_Mechanics.html | |
| Feb 16, 2012 at 18:30 | comment | added | Deane Yang | Good point. But any chance you could provide a few details for those of us who don't have easy access to Abraham and Marsden? | |
| Feb 16, 2012 at 16:49 | history | answered | alvarezpaiva | CC BY-SA 3.0 |