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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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Tautological 1-form on the cotangent bundle

I'm trying to understand a little bit about symplectic geometry, in particular the tautological 1-form on the cotangent bundle. I'm following Ana Canas Da Silva's notes. On page 10 she describes the coordinate free definitions and gives an exercise…
Zorba le Grec
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Rolling out pie dough

I was rolling out a pie crust tonight. I would like to produce a perfect circle, but part way through the border of my crust was a rather different closed curve. I am aware of the Riemann mapping theorem which says I can map my existing crust to a…
Ross Millikan
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Arnold's theorem on action-angles.

I changed the question slightly in its form to make it more readable. I have a question about the action-angle theorem on p. 283 in Arnold's textbook on classical mechanics.(I added the link to this book in the last part of this question) If you…
user167575
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Symplectic reversing diffeomorphisms

Let $(M,\omega)$ be a compact symplectic manifold. Is there always a diffeomorphism $f$ on M with $f^{*}\omega =-\omega$?
Ali Taghavi
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What is symplectic geometry?

EDIT: Much thanks for answers. As was pointed out, the question as it stands is a little too broad. Nevertheless, I don't want to delete it, because I think that such introduction-style questions can be answered without writing a book, rather…
mz71
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Tangent space of Cotangent bundle at zero section?

Let $M$ be a differentiable manifold with cotangent bundle $T^*M$. How can I prove that $T_{(p,0)}T^*M$ is naturally isomorphic to $T_pM\oplus T_pM^*$? If this true, then I think I could prove that the Hessian of $f\colon M\to \mathbb{R}$ is…
Topologieeeee
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Why spheres are not symplectic manifolds?

Reading some books on diferential geometry, a found that $S^{2n}$ (with $ n>1$) are not symplectic manifolds. They say it's because the de Rham cohomology of this spheres are R, but I do not understand this argument. It will be helpfull if anyone…
Edgar Ortiz
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Checking my understanding of $T^*M$ as a symplectic manifold and the links between the classical description of Lagrangians vs this invariant way.

I am working through a book titled "An introduction to mechanics and symmetry" by Marsden and Ratiu. I have written up a brief summary trying to solidify my understanding of the general principles. Two questions: i) Is my understanding of the…
muzzlator
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Why is the moduli space of flat connections a symplectic orbifold?

In her Lectures on Symplectic Geometry on page 159, Ana Cannas da Silva writes "It turns out that $\mathcal{M}$ is a finite-dimensional symplectic orbifold." Can somebody give me a reference for this result, preferably including a detailed…
Daan Michiels
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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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Symplectic form on a complex manifold

I am a little muddled and am hoping I can get some clarification about forms in a complex manifold. Since I am only concerned with local issues, consider $M = \mathbb C^n$ as a complex manifold. So I have complex coordinates $z_1,\ldots,z_n$ and…
Eric O. Korman
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Prove that symplectic Lie algebras, $\mathfrak{sp}(n)$, are simple

The symplectic lie algebra defined by $sp\left(n\right)=\left\{ X\in gl_{2n}\,|\, X^{t}J+JX=0\right\}$ when $J=\begin{pmatrix}0 & I\\ -I & 0\end{pmatrix}$. So $X\in sp\left(n\right)$ is of the sort $X=\begin{pmatrix}A & B\\ C &…
IBS
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The physical meaning of a symplectic form.

So I've studied a bit about symplectic geometry, and I know that phase space is a symplectic manifold, and the symplectic form induces a poisson bracket. However, what is the physical meaning of the symplectic form? Perhaps it's simply the same as…
AkatsukiMaliki
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Physicist trying to understand GIT quotient

I am reading Nakajima's textbook on Hilbert Schemes. I am trying to understand some very basic facts about the GIT quotient. We start with a vector space $V$ over $\mathbb{C}$. Let $G \subset U(V)$ be a Lie group and $G^{\mathbb{C}}$ its…
Marion
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Two different definitions of a Liouville measure

Ok, I'm currently confused because of two different definitions for the Liouville measure associated to a smooth manifold $M$ of dimension $n$. These are: a) The measure $\mu$ on the cotangent bundle $T^*M$ induced by the volume form $(d\alpha)^n$…
DoktorFaustus
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