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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.

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Time-dependent Hamiltonian isotopy and Hamiltonian symplectomorphism

I am having some difficulty understanding the definitions of Hamiltonian isotopy and Hamiltonian symplectomorphism. I know that if we have a Hamiltonian function $H: M \to \mathbb{R}$, we have a Hamiltonian vector field $X_H$ associated with it…
Pedro G. Mattos
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Two Claims about Symplectic Group Actions

There are two claims in McDuff-Salamon's Introduction to Symplectic Topology, 3rd edition on p. 202 that I've been trying to figure out but haven't been able to. Let $G$ be a Lie group acting symplectically on symplectic manifold $(M,\omega)$; this…
inkievoyd
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$1$-form on a symplectic manifold.

If $\omega$ is a $1$-form on a symplectic manifold, will it be closed? It seems to be trivial that if $\sigma$ is symplectic structure on a manifold $M$, then the induced map $$\sigma^\vee: TM\to T^*M$$ is an isomorphism. Hence there exists some…
Junu
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Darboux theorem and symplectomorphisms

In the lecture note I am reading there is following claim: Let $(M,\omega)$ be a symplectic manifold, $f,g : M \rightarrow M$ symplectomorphisms, and $L \subset M$ a Lagrangian. Suppose $f(x) = g(x)$ for all $x \in L$. Then on a neighbourhood of $L$…
Darboux
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Definition of Hamiltonian vector fields

Suppose that $(M,\omega)$ is a a symplectic manifold. Since $\omega$ is non-degenerate, it sets up an isomorphism $$\omega:TM\rightarrow T^*M$$ between $TM$ and $T^*M$. Why does non-degeneracy ($\omega\wedge\omega\wedge\cdots\wedge\omega\neq0$)…
user46348
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About symplectic embedding

I never read about the symplectic embeddings. While reading a general math note, I have following question: Does every symplectic manifold $(M,\omega)$ can be symplectically embedded to some $(\mathbb R^n, \omega_{std})$. Can I have an counter…
Junu
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How do level sets of a time-dependent Hamiltonian function behave under the Hamiltonian flow

Let $(M,\omega)$ be a compact symplectic manifold. Given a Hamiltonian funciton $H:M\rightarrow \mathbb{R}$ one can defined its Hamiltonian flow $\Phi^H_t$ and we will have that given $p\in M$ then $\phi_t^H(p)\subset H^{-1}(p)$ since…
Someone
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Why circle action is not hamiltonian

We have given an action $$S^1\times T^2\to T^2$$ $$(t,(\theta_1,\theta_2))\to (\theta_1+t,\theta_2)$$ Why this action can't be Hamiltonian? May i get some hint, comment suggestion. Thanks.
Junu
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On a Hamiltonian diffeomorphism of the annulus

Suppose that we have the symplectic manifold $(M,\omega)$ where $M$ is a certain annulus, let's say of minimum radius $r_1$ and maximum radius $r_2$, and $\omega=dr\wedge d\theta$. There exists a symplectomorphism $\psi:M\rightarrow M$ that switches…
Someone
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Finding Hamiltonian that gives me a Hamiltonian symplectomorphism in $S^2\times S^2$

Consider $S^2\times S^2$ with its standard symplectic form. I have seen the following statement made that $\Psi(x_1,y_1,z_1,x_2,y_2,z_2)=(-x_1,y_1,-z_1,x_2,-y_2,-z_2)$ is an Hamiltonian symplectomorphism. Well after thinking about this for a while I…
Someone
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Closed curve on oriented surface

Let $ C $ be a closed curve on the oriented field and $ H $ a Hamiltonian. Prove that $ X_{H} $ is somewhere tangent on C.
Yang Ho-Hun
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Symplectomorphism "with a multiplier"

A symplectomorphism is defined as a function $f$ fulfilling $$ f^* \omega = \omega \quad ,$$ where $\omega$ is the canonical form on some even-dimensional space. In particular, the jacobian of any symplectomorphism $ T:(q,p) \mapsto (x,y)$ is a…
QuantumBrick
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Hamiltonian flow of $\pi^* H$ on the graph of $dH$ in a contangent bundle

Let $H:M \to \mathbb{R}$ be a smooth function on a smooth manifold $M$ and consider the pullback $\pi^* H$ of this function to the cotangent bundle $\pi:T^*M \to M$. There is a standard symplectic form on the cotangent bundle $\omega = d \lambda$.…
inkievoyd
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Holomorphic buildings vs. compactification of moduli space of holomorphic curves

Let $(W^{2n}, \Omega)$ be a symplectic cobordism from $(Y_+, \lambda_+, \omega_+)$ to $(Y_-, \lambda_-, \omega_-)$, where $(Y_+, \lambda_+, \omega_+)$ are $(Y_-, \lambda_-, \omega_-)$ are manifolds with stable Hamiltonian structures. Let…
seumath12345
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canonical one-form

The canonical one-form is defined here: http://books.google.nl/books?id=uycWAu1yY2gC&lpg=PA128&dq=canonical%20one%20form%20hamiltonian&pg=PA128#v=onepage&q&f=false My problem is this: It states that if $(x_1,\dots x_n)$ are local coordinates in…
simon
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