Timeline for are there natural examples of classical mechanics that happens on a symplectic manifold that isn't a cotangent bundle?
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| Apr 13, 2017 at 12:58 | history | edited | CommunityBot |
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| Nov 10, 2013 at 22:45 | comment | added | Igor Khavkine | It is also worth emphasizing that it is not always the jet bundle itself that carries the symplectic structure. Rather, the symplectic form is ultimately defined on (the phase space) $\cong$ (equivalence classes of solutions) $\cong$ (equivalence classes of initial data), provided the initial value problem is well posed. The latter space coincides with the fiber of a jet bundle only for ODEs (mechanics without "gauge redundancy"), but not for PDEs (field theory), unless they are cast into the form of infinite dimensional ODEs. | |
| Nov 10, 2013 at 22:40 | comment | added | Igor Khavkine | @symplectomorphic, you're right that jet bundles do not carry a natural symplectic structure. A symplectic structure arises only in conjuction with a Lagrangian (you can find more detail about that by following the link in the first paragraph of the answer, or at ncatlab.org/nlab/show/phase+space#ViaSymplecticStructure). Physically speaking: there is no a priori symplectic structure without dynamics. | |
| Nov 10, 2013 at 19:53 | comment | added | symplectomorphic | Great answer -- but I'm confused. You say the reason we should care about cotangent bundles is that they are special cases of more general symplectic structures. And then you suggest that the relevant general structure is a jet bundle. But jet bundles don't carry natural symplectic structures, do they? Can you tell me where in your story more general symplectic structures appear? | |
| Nov 10, 2013 at 14:40 | history | answered | Igor Khavkine | CC BY-SA 3.0 |