Timeline for What is a symplectic form intuitively?
Current License: CC BY-SA 2.5
9 events
| when toggle format | what | by | license | comment | |
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| Apr 6, 2010 at 13:13 | vote | accept | Jan Weidner | ||
| Apr 6, 2010 at 13:13 | |||||
| Apr 1, 2010 at 17:20 | comment | added | Theo Johnson-Freyd |
@JFO: Certainly I agree that $(M,\omega,H)$ and $(M,\varphi^*\omega,\varphi^*H)$ have the same data. My objection is that there is something about physics that neither one captures. Which is to say, we have to use lossy models all the time, because the universe is much too big to accurately model. In any case, my reason for preferring cotangent-bundle picture is that I spend more of my time doing Lagrangian mechanics, which is inherently about the tangent bundle.
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| Apr 1, 2010 at 7:39 | comment | added | José Figueroa-O'Farrill | Consider a classical physical system described by a triple $(M,\omega,H)$ where $(M,\omega)$ is a symplectic manifold and $H$ is a hamiltonian function. Apply a diffeomorphism (not even a symplectomorphism) $\varphi$ to this data to arrive at $(M,\varphi^*\omega,\varphi^*H)$. I would say that both systems describe equivalent physics, by which I mean the flows of the corresponding hamiltonian vector fields are related by a diffeomorphism. This is not to say that diffeomorphisms are symmetries of a physical system, but neither are symplectomorphisms unless they also preserve the hamiltonian. | |
| Apr 1, 2010 at 7:32 | comment | added | José Figueroa-O'Farrill | I think I'm with Mariano in this one. One of the lessons learnt from the dualities paradigm of the last 15 years or so is that the distinction between "position" and "momenta" (and I use quotes because the statement is much more general) is largely a matter of interpretation. The same <em>physics</em> can be described using different choices of dynamical variables. | |
| Mar 31, 2010 at 19:34 | comment | added | Steve Huntsman | To say that all canonical transformations are symplectomorphisms is not to say that all symplectomorphisms are canonical transformations. | |
| Mar 31, 2010 at 19:10 | comment | added | Theo Johnson-Freyd | @MSA: I just mean that the actual real physical world is (pretty close to) a cotangent bundle, with a specified base. So if I think about it just as a symplectic space, I lose some information. | |
| Mar 31, 2010 at 17:50 | comment | added | Mariano Suárez-Álvarez | Hmm. Why should there be a difference between position and momentum to be preserved by phisical symmetries? | |
| Mar 31, 2010 at 17:19 | comment | added | Jan Weidner | Thanks, thats an interesting viewpoint. What, if you start with a cotangent bundle and do hamiltonian reduction? You end up with a symplectic manifold, that is not a cotangent space anymore, though I would still call it mechanics. | |
| Mar 31, 2010 at 16:27 | history | answered | Theo Johnson-Freyd | CC BY-SA 2.5 |