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.

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Suitable reference for learning symplectic geometry

I am interested in studying symplectic geometry by myself and I'm looking for a good text to use as a reference in the way. I am a bit lost because I've found a lot of notes and books on the subject ...
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KAM theorem for symplectic maps using Generalized Implicit Function Theorems

I've been studying KAM theory for a while and as many of you surely know, there exist many methods in proving "KAM theorems" for different settings. Most of the literature deal with the persistence of ...
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Over the reals the intersection of the orthogonal and symplectic groups in even dimension is isomorphic to the unitary group in the half dimension.

See the answer here. Over the reals the intersection of the orthogonal and symplectic groups in even dimension is isomorphic to the unitary group in the half dimension:$$ U(n) = O(2n, \mathbf{R})...
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Delzant theorem for polyhedra

Delzant theorem says that there is a 1-1 correspondence between compact toric symplectic manifolds (modulo equivariant symplectomorphism) and the Delzant polytopes (modulo lattice isomorphism). The ...
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Computing real de Rham cohomology of Hironaka's 3-manifold example

I have read the construction of Hironaka's famous 3-manifold example: in short, it is a union of two smooth curves $C$ and $D$ in a smooth projective 3-manifold $P$ which intersect each other at two ...
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What does the notation $G\times_P\mathfrak{p}^\perp$ mean, for $P\subset G$ Lie groups?

Suppose $G$ is a Lie group, $P$ a Lie subgroup with $\mathfrak{p}$ the associated Lie algebra. What object is $G\times_P\mathfrak{p}^\perp$? I don't understand what the $\times_P$ means, ...
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Showing Hofer's metric is bi-invariant

Im trying to show that the Hofer metric on a symplectic manifold is bi-invariant but im struggling. Firstly, given the flow $\rho_t$ of a hamiltonian $H_t$, the Hofer metric is $$ d(\rho_0,\rho_1)=\...
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Darboux's Theorem Alternate Proof

I've been given the task of proving Darboux's theorem through non-standard means. Definitions Let $(M,\phi)$ be a symplectic manifold. $\mathcal{F}_{\text{SP}(V)}(M)$ is the bundle of frames $(\eta^...
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How the canonical symplectic form acts

I've read that the canonical symplectic form $\omega$ on $\mathbb R^{2n}$ is given by $$\omega=\sum_{i=1}^n dp_i\wedge dq_i,$$ where $(p_1,\dots,p_n,q_1,\dots,q_n)$ are the coordinates on $\mathbb R^{...
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Passage in a proof from Hofer-Zehnder

The proof I'm referring to is to the following theorem. Assume $S$ is a compact regular and strictly convex energy surface for the Hamiltonian field $X_H$ in $\mathbb{R}^{2n}$. Then $S$ carries a ...
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Is this statement on symplectic maps completely general or does it need some extra hypotheses (as the ones with which I proved it)?

A lemma from McDuff-Salamon says that $\psi:\mathbb{R}^{2n}\to\mathbb{R}^{2n}$ is symplectic iff $\{F,G\}\circ\psi=\{F\circ\psi,G\circ\psi\}$. I proved that. Then there is an exercise showing that $\{...
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Showing that the intersection of two particular vector spaces has codimension 1 in the smaller of the two spaces

Here is the setting: I have a compact symplectic manifold $(X^{2n},\omega)$ and a compact symplectically embedded submanifold $(M^{2d},\sigma)$; that is, $\iota^{\ast}\omega=\sigma$. The dimension 2d ...
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57 views

Corresponding toric variety for n-simplex

Let $P $ be a Delzant polytope and $X_P $ be a corresponding Toric variety. I want to see if $P=\sum $ be a n-simplex then $X_P=\mathbb P^n$
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Computing the signature of the intersection form on the middle cohomology of compact, symplectic, non-Kaehler manifolds…

For a compact Kaehler manifold, one can compute the signature of the intersection form on the middle-degree cohomology, by taking an alternating sum of the Hodge numbers (this is the Hodge Index ...
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construct a connection such that a given tensor is parallel wrt it

Let $\omega$ be a symplectic form on a smooth manifold $M$. How does one construct a connection on $TM$ such that $\omega$ is parallel to it? It's easy to construct a connection on a dual bundle ...
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34 views

Does $\{f,g\}$ mean anything when neither $f,g$ are the hamiltonian of a system?

Say one has a mechanical system with hamiltonian $H$, and two other arbitrary observables $f,g$. $H$ is super useful since $\{H, \cdot\} = \frac{d}{dt}$. But does $\{f,g\}$ give any useful information ...
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Geometric interpretation of kernel and critical points of a moment map

A moment map $\mu$ is defined when one has a Hamiltonian $G$-action on a symplectic manifold $M$, for some Lie group $G$. My question is, what are the geometric interpretations of the kernel and ...
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What is the actual definition of “polarizing form”, in the context of cohomology algebras?

I am studying Voisin's Hodge Theory and Complex Algebraic Geometry, in order to better understand the underpinnings of her 2008 paper on Hodge structures. She discusses elements $\omega$ in $H^{2}(...
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Intuition behind variational principle

Hofer-Zehnder, in section 1.5, proves that every Hamiltonian field on a strictly convex compact regular energy surface carries a periodic orbit. I have understood the proof. What I am wondering about ...
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When is a vector field hamiltonian with respect to some symplectic form?

Given a vector field $v$ on a $2n$-dimensional manifold, how many symplectic forms are there on $M$ that make $v$ a hamiltonian vector field? Alternatively, take the set of all $(H,\omega)$ pairs, mod ...
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Canonical Transformation and Symplectic Conditions

I have one question regarding Canonical transformation and symplectic matrix. I have read some notions from the following note: http://www.chim.unifi.it/orac/MAN/node6.html For me it is not clear ...
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Poincaré-Birkhoff theorem in sympl. geometry

On p. 274 of McDuff and Salamon's Introduction to symplectic topology a corollary to the Poincaré Birkhoff theorem is presented. So we are given an area preserving map on an annulus $\psi(x,y)=(f(x,y)...
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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 ...
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What is the general form of an area-preserving map of $\mathbb{R}_2$?

A Hamiltonian flow generates such a map. But what is the most general form of such a map? Any theorem? Any procedure to generate such a map? Of course, I want the map to be continuous.
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Exterior 2-form, 1-form, Hodge star operator.

In $\mathbb{R}^{2n}$ with coordinates $x_1, x_2, \dots, x_{2n}$, consider an exterior 2-form$$\eta = \sum_{k=1}^n x_{2k-1} \wedge x_{2k}.$$Given a 1-form $\alpha = \sum_{i=1}^{2n} a_ix_i$, what is the ...
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Formula about time derivative of pushforward of family of forms: where is it from?

Proving Darboux's theorem, Hofer-Zehnder try to find, given $\omega$ a closed nondegenerate 2-form and $\omega_0$ the canonical symplectic form, a family of diffeomorphisms $\phi^t$ such that for all $...
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rigid body poisson bracket

i have trouble to understand the definition of the rigid body poisson brackets. In the book of Marsden and Ratiu "Introduction to Mechanics and Symmetry", in chapter 10.1 they introduce the poisson ...
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Is the induced volume on submanifolds unique?

consider a $2n$-dimensional manifold $\mathcal{M}$ With a volume element $\omega$. Now consider a $(2n-2)$-dimensional submanifold $\mathcal{N}$. How one can define a volume on $\mathcal{N}$ based on $...
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smooth manifolds and preimage of momentum mappings

Let $G$ be a Lie group acting on itself as $\phi(h)(g)= L_h(g)$ as a left translation. Then we can consider the cotangent lift of this action, namely $\Phi: G \times T^*G \rightarrow T^*G$ as $\Phi(h)...
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Determinant structure of symplectic matrix

I want to show that if $\lambda$ is a real eigenvalue of a symplectic matrix $A$ then its char poly is of the form $\det(A-\mu id) = (\lambda-\mu)(\frac{1}{\lambda}-\mu) \det(\hat{A}- \mu id) $ where $...
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Definition of $k$-precosymplectic manifold

A precosymplectic manifold of rank $2r$ is a triple $(M,\omega,\eta)$ where $M$ is a smooth manifold of dimension $2m+1$, $\omega$ is a closed 2-form on $M$ and $\eta$ is a closed 1-form on $M$ such ...
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Determining the corresponding vector field to a group action.

Im having trouble trying to understand how to determine the corresponding vector field to a group action on a symplectic manifold. I feel this will be easier if I give two examples which are confusing ...
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Symplectic transform between pairs of Lagrangian subspaces

Let $V$ be a symplectic space. Suppose $(W_1, W_2)$ and $(U_1, U_2)$ are pairs of complementary Lagrangian subspaces. There exists a symplectic transform which maps $W_i$ into $U_i$, right? There ...
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Symplectic form and wedge sum

The wedge sum of $k$ symplectic 2-forms is given by ( if $\omega = \sum_i e_i^* \wedge f_i^*$) $$ \omega^k = k! \sum_{1\le i_1 <...<i_k \le n} (e_{i_1}^* \wedge f_{i_1}^*) \wedge ... \wedge (...
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Using the flow to show a certain compact connected set isomorphic to a torus.

So i've found this lemma in some notes on symplectic manifolds and I think i'm lacking some basic knowledge on flows as I the hint for the proof makes no sense to me. It goes as follows. Lemma 18.11: ...
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Integrate symplectic two-form

I am supposed to show the following (maybe falsely stated) theorem Let $\Phi$ be a symplectic group action $\Phi: G \times M \rightarrow M$ on a manifold $M$. Assume that the symplectic form $\omega$ ...
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Lie group action from the Lie algebra

want to find the corresponding lifting f the standard U(n) lifting on $C^n$ to $L=C^n \times C$ with hermitian metric $e^(-|z|^2)$. I try to follow the method in Donaldson, and I find if B in u(n) ...
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Typo in “Intro to Contact Topology” by Geiges, Lemma 1.4.10?

In Introduction to Contact Topology by Geiges, there is a result relating Hamiltonian and Reeb flows for hypersurfaces of contact type in a symplectic manifold. Lemma 1.4.10 $\,$If a codimension 1 ...
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Without loss of generality in proof about subspaces in symplectic linear algebra

A linear symplectic space is a 2n-dimensional vector space $V$ with a symplectic two form $\omega.$ On this vector space $V$ is a canonical basis $(e_1,...,e_n,f_1,...f_n)$ with $\omega(e_i,f_j) = \...
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Problems understanding this proof

$\textbf{Theorem 3.5.10}$ (Arnol'd $[1]$). Suppose $(M,\sigma)$ is a symplectic manifold of dimension $2n$, let $f_1,...,f_n$ be an involution on $M$, and finally assume that the Hamilton fields $H_{...
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Trouble understanding proof of this proposition on contact type hypersurfaces

I have finally started working on my thesis. I am reading the first reference book, which is Dusa McDuff, D. Salamon, Introduction to Symplectic Topology. It's a tough struggle, given my not-too-great ...
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Mistake in book on symplectic topology?

I just read the proof of the non-squeezing theorem in "Introduction to symplectic topology" by Mc Duff and Salamon. The thing that is strange is that they say: Let $\Psi$ be the linear transform ...
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Two definitions of conormal bundle

Suppose $X$ is a smooth manifold and $f: Z \to X$ is an immersion with transverse self-intersection. I've seen (e.g. here) the conormal bundle to $Z$ in $T^* X$ defined as Definition A: $\quad L_Z := ...
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Question on symplectic geometry

I am currently reading this paper on symplectic geometry. It deals with the question how the stability properties of a sequence of (periodic) points or fixed points can be related to the second ...
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Question for experts in dynamical systems or symplectic geometry

I am looking for a theorem (read it once, but forgot about it and would now like to find a reference with proof): In symplectic geometry (at least for some particular subset of $\mathbb{R}^2$), there ...
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Easy classical physics made mathematically rigorous!

Consider the following: We are given a symplectic manifold $M$. Now, we define a Hamilton function $H : M \rightarrow \mathbb{R}.$ Additionally, we want that $H^{-1}(x)=:M_x$ is a submanifold. We can ...
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Gaussian unitary dilation of Gaussian channels

I am starting with a few definitions. All these are standard and can be accessed from some quantum information or quantum physics books, for instance the books by Holevo or Parthasarathy. The question ...
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Symplectic manifold

Let ($M$, $\omega$) be a symplectic manifold of dimension $2n$. Then $\omega$ is non-degenerate $2-form$ by definition. Now, my question is if we can conclude that $\omega \wedge ...\wedge \omega$ ...
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Hamiltonian flow local diffeomorphism?

I am currently reading Arnold's proof of the Darboux theorem in his book on classical mechanics and fail to understand some point. The background So he wants to show that any symplectic form is ...
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115 views

Exterior derivative

Let $$\omega = \frac{1}{2} \sum_{i,j} \omega_{i,j} dx_i \wedge dx_j$$ be an antisymmetric form, so i.e. $\omega_{i,j} = - \omega_{j,i}.$ Now, in some lecture notes I found that $$d \omega (X_F,X_G,...