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Results tagged with sg.symplectic-geometry
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user 12482
Hamiltonian systems, symplectic flows, classical integrable systems
1
vote
model compact coisotropic submanifold
There are well-known normal form theorems for all kind of constant rank submanifolds in symplectic manifolds, even global statements (and only such a thing seems to be of interest if you want to talk …
- 8,098
6
votes
Why can we define the moment map in this way (i.e. why is this form exact)?
There is yet another interpretation of the already mentioned obstructions for the existence and uniqueness of momentum maps (moment mappings, moment maps, momentum mappings....) in terms of Poisson ge …
- 8,098
41
votes
What is a Lagrangian submanifold intuitively?
Everything is a lagrangian submanifold (A. Weinstein's lagrangian creed...)
Indeed, every manifold is the lagrangian zero section of its cotangent bundle ;)
But more serious: Weinstein's tubular neig …
- 8,098
8
votes
Accepted
Lagrangian Submanifolds in Deformation Quantization
Well, in the symplectic case, the situation is somehow much simpler as in the general Poisson case where you only can speak about coisotropic (there is no good meaning of minimal coisotropic as the ra …
- 8,098
3
votes
Can you tell the volume of a symplectic manifold from the Poisson brackets?
Let me add a few remarks on Theo's answer. For a compact connected symplectic manifold, it is known that the integration with respect to the Liouville volume form (whatever normaliyation you prefer) i …
- 8,098
12
votes
Why are Lagrangian subspaces in a symplectic vector space interesting?
In symplectic linear algebra this is perhaps not completely clear. However, if one passes to symplectic geometry then the Lagrangean submanifolds indeed play a dominant role. This is in some sense Wei …
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18
votes
Applications of symplectic geometry to classical mechanics
The list will be long, very long indeed. But to start:
Questions about dynamics of Hamiltonian systems are at the heart of symplectic topology, symplectic capacities are precisely introduced for that …
- 8,098
4
votes
Accepted
Open symplectic embeddings and deformation quantization
Hi Igor,
there is a quite elementary way to see that star products restrict to open subsets: it's essentially part of the definition of a star product. Here, I will focus on the case of smooth (sympl …
- 8,098
4
votes
Accepted
Prequantization and Hilbert space
OK, so here are just a few thought on this large topic of quantization. First of all, the question of irreducibility can equally well be asked for deformation quantization (as mentioned by other answe …
- 8,098
8
votes
Accepted
In the dictionary between Poisson and Quantum, what corresponds to Coisotropic?
Concerning your first question I have a couple of suggestions: first coisotropic is in some sense the best we can have in a truely Poisson situation: there is nothing like lagrangian (unfortunately).
…
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78
votes
The Planck constant for mathematicians
Let's give it a try. Of course, the precise mathematical meaning is perhaps absent, so the answers are sort of heuristic. But if I understand correctly, you want to gain intuition ;)
The first observ …
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