Timeline for The Lagrangian formulation of mechanics without going through variational principles.
Current License: CC BY-SA 3.0
8 events
| when toggle format | what | by | license | comment | |
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| Sep 30, 2013 at 13:35 | vote | accept | agt | ||
| Nov 25, 2011 at 17:48 | comment | added | Igor Khavkine | Thanks! I should point out that the bulk of the nLab phase space page was written by Urs Schreiber, as can be seen from the revision history: ncatlab.org/nlab/history/phase+space | |
| Nov 25, 2011 at 16:28 | comment | added | agt | Dear Igor Khavkine, your answer is fantastic, and even more is such your work at the nLab about the phase space. This is much more than I would have expected, thank you. | |
| Nov 25, 2011 at 14:14 | comment | added | Spiro Karigiannis | This is a great answer! Thanks very much. | |
| Nov 25, 2011 at 14:14 | comment | added | user17945 | Oh, ok, I see now what you're saying. I didn't realise you were starting with the symplectic form defined by the Lagrangian (although I should have). Thanks. | |
| Nov 25, 2011 at 10:28 | comment | added | Igor Khavkine | Consider the phase space coordinatized by initial data at time $t=0$, say for 1-dimensional particle motion. Then natural coordinates are $(x,\dot{x})$. In these coordinates the symplectic form will look like $\omega=\omega(x,\dot{x}) dx\wedge d\dot{x}$. The Legendre transform defines $p=p(x,\dot{x})$. The new coordinate system $(x,p)$ is special because $\omega=dp\wedge dq$ is now in canonical form, while in the $(x,\dot{x})$ coordinates it is not. Darboux's theorem guarantees that such special coordinates always exist (locally), but they need not always be easy to find. Here they are. | |
| Nov 25, 2011 at 8:59 | comment | added | user17945 | "Any symplectic manifold has local coordinates in which the symplectic form is canonical (Darboux's theorem). The Legendre transform identifies this choice of coordinates explicitly." Sorry, can you clarify what you mean by this? I'm not seeing the connection between the Legendre transform and canonical coordinates. | |
| Nov 25, 2011 at 4:10 | history | answered | Igor Khavkine | CC BY-SA 3.0 |