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Questions tagged [symplectic-geometry]

Symplectic geometry is a branch of differential geometry and differential topology which studies symplectic manifolds; that is, differentiable manifolds equipped with a closed, nondegenerate 2-form.

Symplectic geometry is a branch of differential geometry and differential topology which studies symplectic manifolds; that is, differentiable manifolds equipped with a closed, nondegenerate 2-form. Symplectic geometry has its origins in the Hamiltonian formulation of classical mechanics where the phase space of certain classical systems takes on the structure of a symplectic manifold. For more information please consult the Wikipedia page on symplectic geometry.

1282 questions
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Why should a symplectic form be closed?

Thanks for reading my question. I'm wonder why a symplectic form should be closed. I found many different answers in the internet, but it sounds like a technical requirement (if we omit this requisit, we obtain almost symplectic structures,…
juliho
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Hamiltonian flows are symplectic

I want to show, in coordinates $(x,\xi)\in T^*\mathbb{R}$, that the Hamiltonian flow $\Phi_t = \exp(t H_p)$ is symplectic for each $t$. Here, $H_p$ is the Hamiltonian vector field determined by the smooth function $p(x,\xi)$. We know that…
Ron
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Is the Poisson tensor associated to a left invariant symplectic form, also left invariant?

Given a left invariant symplectic form $\omega$ on a Lie group $G$, the Poisson tensor associated to $\omega$ is given by $$\pi(df,dg)=\omega(X_f,X_g)$$ where $X_f$ is the hamiltonian vector field associated to the function $f$ ;…
amine
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symplectic manifolds

I know sometimes they use advanced methods to prove a given 4-manifold is not symplectic. for instance by Seiberg-Witten theory. But for a manifold to be symplectic we just need to check that there is an element in the second cohomology which is…
7779052
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Proof that $S^4$ is not a symplectic manifold

I started learning symplectic geometry, but apparently, i forgot some of my differential geometry. I am studying from the book by Aebischer et al. In it, it is claimed that $S^4$ is not a symplectic manifold, and the proof goes as follows: suppose…
guest
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Understanding Monodromy by examples

What is the intuition behind "monodromy"? Could you explain with some examples? For instance, what does it mean "monodromy around a singular fiber is a dehn twist" I don't understand what it means pictorially.
contakto
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Hyperbolic lagrangian in a symplectic $6$-manifold

Is there an example of a closed symplectic $6$-manifold and a closed lagrangian sub-manifold, which is diffeomorphic to a hyperbolic $3$-manifold?
Nick L
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the tautological 1 form

My question relates to p317-p318 of John Lee's "Introduction to Smooth Manifolds" discussion about the tautological 1 form. In Proposition 12.24, we have the expression: $\tau_{(x, \xi)} = \pi^* (\xi_i dx^i) = \xi_i dx^i $ Here $\tau_{(x, \xi)} \in…
codethink
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Second Hirzebruch surface as Delzant space associated to trapezoid

I am trying to understand how the second Hirzebruch surface arises as the Delzant space associated to the trapezoid $\Delta \in (\mathbb{R}^2)^\ast$ given by the vertices $(0,0) , (1,0), (1-a,a), (0,a)$. Applying the Delzant construction to the…
nora
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Passage in proof existence of Symplectic structure on Symplectic fibration

I need help in understanding the last passage in Thurston’s proof of the existence of a compatible Symplectic form on a Symplectic fibration $M\to B$ With Symplectic base space $(B,\beta)$. More precisely the setting is the following. Let…
Luigi M
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Finding a path of symplectic forms

Suppose $M$ is a symplectic manifold and $\omega_0$ and $\omega_1$ are two symplectic forms belonging to the same de Rham cohomology class. Assume in addition they can be connected via a path of symplectic forms lying in this same class. Suppose…
user888
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Reduction along an Orbit for C.-M. systems

I am having trouble in understanding the section of this paper http://www-math.mit.edu/~etingof/zlecnew.pdf where the author introduces the Calogero-Moser system as the reduction of a manifold $M$ on which is acting a group $G$ along the coadjoint…
Brightsun
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Quantitative question on almost symplectic structures.

A non-degenerate 2-form $ \omega $ on a manifold $X$ is called an almost symplectic form. A manifold possesses such a form if and only if it is almost complex. One doesn't necessarily have a closed symplectic form however, for example, the…
Yalim O
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Poisson Bracket composed with a symplectomorphism

I'm trying to understand the proof of Lemma 5.2.1 in McDuff-Salamon (3rd edition). Let $(M, \omega)$ be a symplectic manifold, and $G$ a Lie group with a symplectic action on $(M, \omega)$. Let $X_\xi, X_\eta$ be the vector fields corresponding to…
Rei Henigman
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Restriction of symplectic form on submanifold

Consider a symplectic manifold $(M,\omega)$, for a 2 dimensional submanifold $N \subset M$, do we always have $\omega|_{N}$ gives a volume form on $N$?If not, is there any condition we can put on $N$ to make this happen?
user388493
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