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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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Exact Deformation of Lagrangian Submanifolds

Let $j_{t}:L \rightarrow P$ be a family of Lagrangian submanifolds. I'm trying to show that the form $j_{t}^{*}(i(X_{t})\omega)$, $X_{t}(j_{t}(x)):=\frac{dj_{t}(x)}{dt}$ is exact for all $t$ if and only if there is a family of Hamiltonian…
Steppewolf
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The submanifold $S^2$ in the symplectic product $S^2\times S^2$

Let $S^2$ come equipped with the usual symplectic form and $S^2\times S^2$ come equipped with the product symplectic form and coordinates $(x,y)$ with $x\in S^2$. Consider the "diagonal sphere" $(x,x)$. Is the sphere symplectic? I feel like it is as…
confusedmathstudent
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Classification of symplectic surfaces and confusion about "symplectomorphism"

So I read that the unique invariant of symplectic surfaces is the total area, i.e. two surfaces are symplectomorphic iff their area is the same. Consider $S^2$ with polar coordinates $(h,\theta)$ wherever these exist and the symplectic forms…
confusedmathstudent
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Does a Hamiltonian-preserving, symplectic vector field provide an integral of motion?

Consider the hamiltonian system $(M, \omega, H)$ with hamiltonian vector field $X$ defined by $$ \tag{1}\label{1} \iota_{X}\omega = -dH $$ It is a simple matter of applying definitions to show that a smooth function $f$ on $M$ is an integral of…
DavideL
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Prove that $S^2$ is symplectic manifold

How to show that $S^2$ is a symplectic manifold? What is the symplectic structure on $S^2$?
Uncool
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Vector fields generating the vector bundle $(TV)^{\bot}$ where $V$ is a submanifold of a cotangent bundle $T*X$

Let $V$ be a submanifold of the cotangent bundle $T^*X$ of a smooth manifold $X$. Then we can consider the vector bundle $(TV)^{\bot}$ whose fiber is the symplectic orthogonal complement of a tangent space of $V$. In some references, there is the…
SoYu
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Square root of Symplectic and Positive Definite Matrices in $M_{2n\times 2n}(\mathbb{R})$

Let $R$ be a symplectic matrix. Then the adjoint $R^*$ of $R$ is also a symplectic matrix. Then $RR^*$ is symplectic and positive definite. I want to know that $(RR^*)^{1/2}$ is also symplectic.
SoYu
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Hamiltonian symplectomorphism

I have some difficult of dealing with time dependent vector fields and I am studying hamiltonian symplectomorphisms. I have two questions 1) If $\psi_t$ is a Hamiltonian isotopy genereted by $H_t$ and $\phi_t$ is a hamiltonian isotopy generated by…
Jude
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Hamiltonian Diffeomorphism

I need to study Hamiltonian diffeomorphism, their definition and structure as a group. Could someone give me a bibliography suggestion other than Hofer Zehnder book? Thank you very much for the help.
Jude
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Question about symplectic tranformations

Suppose I know that two vectors $\vec{a}$ and $\vec{b}$ are perpendicular in a given basis spanned by basis vectors $\vec{x}$. Now suppose I transform to another basis $\vec{x'}$ using a symplectic transformation matrix S (i.e. $SJS^{T} = J$ for…
user34801
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Integral of motion for a Hamiltonian G-space (M, ω, G, μ)

I am reading ana cannas' lecture notes to try understanding a bit about symplectic geometry. At page 147, there is this definition : Definition 24.2 A G-invariant function f : M → R is called an integral of motion of (M, ω, G, μ). If μ is…
roi_saumon
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Recommend one good Symplectic topology textbook

I need to gain some idea of this topic (and holomorphic curves) by the end of next semester. So, if you can, please suggest a textbook or some lecture notes that'll help to build geometric insights.
nandi
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Standard Symplectic Form

I am stuck at how to make the matrix of standard symplectic form. The given conditions are Definition Let $V$ be a vector space over $\mathbb{R}$. Then $\omega\in \Lambda^2(V)$ is called symplectic form on $V$ if $\omega' : V\rightarrow V^*$…
Lev Bahn
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Is the exponential map a symplectomorphism?

In the case of $\mathbb{R}^{2n}$ the exponential map just becomes $$ exp_{p}(v) = p + v$$ which results in $$ Dexp_{p}(w)_{v} = w $$ and proves that this map is a symplectomorphism. I am wondering if this is also the case in a general symplectic…
Netivolu
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Find explicit symplectomorphism

How to solve the following: Find explicit symplectomorphism between (2-sphere of radius r without one point) and (open disc of radius 2r)? I don't have any idea how to solve this, so any help is welcome. Thanks in advance.
alans
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