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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.

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Showing existence of symplectic transformations preserving a quadratic form

Question: I need help to prove the following statement. Let $W_i:=w_iw_i^T\in\mathbb{R}^{n\times n}$, for $n$ even, be symmetric rank-1 matrices, $J=-J^T$ the canonical symplectic matrix and define ...
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Bounding norms of symplectic matrix factorisations and non-seperable Hamiltonian flows

Problem setup: Let $e^{hJM}$ be the time-$h$ flow corresponding to the ODE $\dot{x} = JMx$, with $M = \left(\begin{array}{cc} A & C\\ C^T & B\\ \end{array}\right)$ symmetric positive ...
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Volume preserving transformation in the Circular Restricted Three-Body problem

the Lagrangian of the planar circular restricted three-body problem in the rotating coordinate frame is: $\mathcal{L}(x,y,\dot{x},\dot{y})=\frac{1}{2}(\dot{x}-\Omega y)^2 + \frac{1}{2}(\dot{y}+\Omega ...
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How to make Lagrangians Transversal by Hamiltonian Perturbations?

Let $L_1$ and $L_2$ be two Lagrangians in a symplectic manifold $M$. In order to define $HF(L_1,L_2)$ (Lagrangian Floer homology of $L_1$ and $L_2$), we need the two Lagrangians to intersect ...
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Understanding the Relationship Between the Principal Symbol of $-\Delta$ and $\sqrt{-\Delta}$ and Geodesic Flow in Hamiltonian Systems

In the context of Hamiltonian systems in symplectic and Riemannian geometry, consider the following fact: Let $(M,g)$ be a Riemannian manifold and $(M,\omega,H)$ a Hamiltonian system with $$H(q,p)=\...
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Representation/factorising of symplectic groups elements

According to Hall Chap. 3 Corollary 3.47: for a connected matrix Lie group $G$, every element $A\in G$ can be written in the form $A=e^{X_1}e^{X_2}...e^{X_k}$ for some $X_i\in g$, where $g$ is the Lie ...
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Principal Symbol of the Fractional Laplacian on Manifolds

In the euclidean space $\mathbb{R}^n$, we can define the Fractional Laplacian as $$(-\Delta)^s f := \int |\xi|^{2s}\hat{f}(\xi)e^{ix\cdot\xi}d\xi.$$ The principal symbol is clearly $p(x,\xi)=|\xi|^{2s}...
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Local behaviour of the action of a finite subgroup of the automorphism group of a Riemann Surface fixing a point

Consider $(\Sigma,j)$ a closed Riemann surface, and $G \subset Aut(\Sigma,j)$ a finite subgroup that fixes a point $z_0 \in \Sigma$. How can one construct a $G$-invariant neighbourhood $U \ni z_0$ ...
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Why $\theta ([\xi_f, \xi_g])=\frac{1}{2}\xi_f \langle \theta ,\xi_g\rangle -\frac{1}{2}\xi_g \langle \theta ,\xi_f\rangle$?

Let $(M,\omega)$ be a symplextic manifold with $\omega =-d\theta$. For $f,g\in C^{\infty}(M)$, we can define a Poisson bracket $\{ f,g\}=\omega (\xi_{f},\xi_f)$ where $i_{\xi_f}\omega=\omega(\xi_f,\...
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Hamiltonian Group Actions on Calabi-Yau Cones

Let $(M, g, J, \omega, \Omega)$ be a Calabi-Yau cone (where $\Omega \in \Gamma(K_M)$ is the parallel holomorphic volume form), and assume we have a Hamiltonian group action $G \circlearrowright M$ ...
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Why $\mathcal{L}_{\xi_X}\omega =0$ when the action is symplectic?

Let $(M,\omega)$ be a symplectic manifold and $G\times M\to M$ a symplectic action on $M$, i.e., each map $M\to M$ by $q\mapsto f\cdot q$ is a symplecticmorphism. If $\xi_X$ is the infintesimal ...
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Momentum map for coadjoint orbit

Let $G\times \mathfrak{g}^*\to \mathfrak{g}^*$ be the coadjoint action defined by $f\cdot q=Ad^*_fq$ where $Ad:G\to GL(\mathfrak{g})$ is the adjoint representation of $G$ with dual $Ad^*$. For $p\in \...
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A question on geometric quantization

Assume $(M,\omega)$ is a symplectic manifold. Consider $H$ to be the space of complex wavefunctions on $M$, $\{ \psi: M\to \mathbb{C}\}$ with scalar product given by $\langle \psi |\phi \rangle =\int_{...
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The components of a two-form form an invertible matrix

Let $M$ be a manifold. A symplectic form $\omega$ on $M$ is a closed, non-degenerate two-form on $M$. Non-degeneracy means that for all $p\in M$, any vector $v\in T_p M$ such that $\omega_p (v,w)=0$ ...
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Complex manifold structure on moduli space of holomorphic bundles on Riemann surface

Consider the space $\mathcal N$ of rank $r$, degree $d$ holomorphic bundles on a compact Riemann surface $X$ of genus $g\geq 2$. Neitzke constructs this by considering $L^2_k$-completions and taking ...
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Why $S^4$ is not symplectic; while $S^2$ is?

Whenever mathematicians try to prove $(S^{2n}, \omega)$ for $n > 1$ is not symplectic, they always invoke the Stokes' theorem and rely on the fact that $\partial S^{2n}=\varnothing$. I understand ...
iliTheFallen's user avatar
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Action-angle variable construction and the assumption of independent actions

Let $N$ be an $n$-dimensional smooth manifold, $M=T^*N$, and $\omega$ its canonical symplectic structure. Any $H\in\mathcal{C}^\infty(M)$ induces a Hamiltonian vector field $X_H$ defined implicitly by ...
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How does an intersection survive through (generic) perturbation?

I am looking for the proof of a folklore statement which I know (or heavily suspect) to be true, but haven't been able to find written down yet. I have a (symplectic) manifold $M$ of dimension $2n$, ...
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What does $\xi_{\mathcal{F}} -\langle \theta ,\xi_{\mathcal{F}}\rangle+\mathcal{F}$ mean?

Assume that $\omega$ is symplectic two-from on a Manifold $M$ such that $\omega =-d\theta$ where $\theta$ is a one-from on $M$. We know that for every $\mathcal{F}\in C^{\infty}(M)$, there is a vector ...
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Area 2-form and index 1-form on the Hyperbolic Semiplane

I'm having some trouble in defining/deducing a natural 2-form of area in the upper-half plane model of the hyperbolic space $\mathbb{H}^2$. Probably this is a very basic question. I set the context ...
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Prove that there does not exists a Symplectic form in $S^3\times T^3$

Let $S^3\subset \mathbb C^2$ be the 3-sphere and $T^3=S^1\times S^1\times S^1$. I am trying to show that there does not exist a symplectic form $\omega$ in $S^3\times T^3$. As far as I know, the ...
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Dimension of the space of primitive forms

Consider $E := \mathbb R^{2n}$ with its standard symplectic form $\omega$, almost complex structure $J$ and metric $g$. The operator "wedge with omega" $L : \Lambda_\mathbb C^* E \to \...
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Show that $\epsilon$ is a regular value for $\tilde{\mu}$ and that the circle action on $\tilde{\mu}^{-1}(\epsilon)$ is free

Assume we have a Hamiltonian action of $S^1$ on $(M, ω)$ with moment map $\mu_M : M → \Bbb R $and consider the action of $S^1$ on $(\Bbb R^2, ω_0)$ by rotation which has moment map $\mu_{\Bbb R_2} : \...
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Hamiltonians on compact surface

Let $\Sigma$ be a compact, connected surface with a non-empty boundary, equipped with an area form $\omega$. Consider a time-dependent, 1-periodic Hamiltonian $H \in C^{\infty}(\mathbb{R}/\mathbb{Z} \...
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computing fixed points of a toric action

Let $W_1=\{ ([a : b], [x : y : z]) \in \Bbb{CP^1} \times \Bbb{CP^2}: ay=bx \}$ Given the action of $\Bbb T^2$ on $W_1$ defined as $(u, v) \cdot ([a : b], [x : y : z]) = ([ua : b], [ux : y : vz])$ How ...
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Darboux chart compatible with symplectic submanifold

Suppose $(M, \omega)$ be a symplectic manifold, and let $\iota : N \hookrightarrow M$ be a symplectic submanifold. Then, are there exists a coordinate system $(U,(z, w))$ covering $N$ such that $w=0$ ...
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Prove that the action of $\Bbb T^2$ on the Hirzebruch surface is Hamiltonian with the given momentum map

I am reading this thesis https://www.few.vu.nl/~trt800/theses/haroldnieuwboer.pdf and on pag 11 there is Example 2.2.3 that I am trying to figure out. Let $W_K=\{ ([a : b], [x : y : z]) \in \Bbb{CP^...
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Find a Lagrangian subspace complementary to two subspaces

Let $(V,\omega)$ be a symplectic vector space and $A,B$ be two subspaces of $V$ with $2\dim A=2\dim B=\dim V$. I need to prove that there always exists a Lagrangian subspace $L$ of $(V,\omega)$ that ...
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Tautological one-form on a product manifold

Let $M$ be a smooth manifold and $\theta$ be the tautological one-form on the cotangent bundle $X := T^*M$ of $M$. Using local coordinates $(x, \xi)$ on $X$, it takes the form $\theta = \xi_i dx^i$. I ...
zarathustra's user avatar
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Moment on manifold, no dependence on Lagrangian anymore?

I am a bit confused about the mathematics of classical mechanics. When working in $\mathbb{R}^n$, the generalized momentum is defined as the derivative of the Lagrangian with respect to the velocity ...
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Stuck on Differential Geometry proof

My concrete questions are (for context see below): Is is true that $i$ as below embeds $M$ in $T^*\mathbb{R}^n$? Is it true that $M$ is Lagrangian in $\mathbb{C}^n$ if and only if $i(M)$ is ...
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Why can we interpret forces on particles as components of a smooth convector field in the n-Body problem?

I'm trying to understand Example 22.18 (The $n$-Body Problem) in John Lee's Smooth Manifolds textbook. I'm confused by the step where the forces in Newton's second law [Eq. (22.12)] are interpreted as ...
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Existence compatible integrable complex structure

Let $(M,\omega)$ be a fixed symplectic manifold. There are lots of compatible almost complex structures, but is there a criterion to decide if any of them is integrable? For example take $\mathbb{R}^{...
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Is there a vector field $X$ on $M$ such that $\omega (X,\cdot )=df$?

Let $(M,\omega)$ be a symplectic manifold and $f\in C^{\infty}(M)$. Is there a vector field $X$ on $M$ such that $\omega (X,\cdot )=df$? where $d$ denotes the exterior derivative. By $\omega (X,\cdot )...
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showing that $\partial^2=0$ in the nerve of a lie groupoid.

I want t prove that $\partial^2=0$ where the map $\partial$ is defined for every $\omega \in \Omega^{p}(\mathcal{G}_p)$ as $$\partial(\omega) = \sum_{i=1}^{p+2} (-1)^i (\partial_i)^* \omega \in \Omega^...
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showing that a 2-form is multiplicative

I want to proof that that the 2-form $\omega_{(g,x)}\in \Omega^2(G\ltimes G)$ guven by $$\omega_{(g,x)} = -\frac{1}{2} [\langle \text{Ad}_x \text{pr}^*_1 \theta_L, \text{pr}^*_1 \theta_L \rangle + \...
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If the almost complex structure $I$(compatible with the metric of a Riemannian manifold) is integrable and $\omega$ is closed then they are parallel

$(M, g)$ : Riemannian manifold, $ I $ : an almost complex structure compatible with $g$ and $\omega $ the corresponding 2-form (that is, $ω^\flat = g^\flat \circ I$). Let $\nabla$ be the Levi-Civita ...
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Understanding an integrable almost complex structure

My notes only say that "Definition An almost complex structure is integrable if it is induced by an underlying complex structure." How do I translate this into a formula? I am not sure why ...
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$I$ is parallel with respect to the Levi-Civita connection if and only if $\omega $ is parallel. $I:$almost non-complex structure

Let $(M, g)$ be a Riemannian manifold, let $ I $ be an almost complex structure compatible with $g$ and $\omega $ the corresponding 2-form (that is, $ω^\flat = g^\flat \circ I$). Let $\nabla$ be the ...
some_math_guy's user avatar
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Does the frame of a direct sum of subspaces split somehow?

Let $V$ be an $n$-dimensional vector space, and let $M\subset V\oplus V$ be an $n$-dimensional subspace. It is known that I can identify $M$ (non-uniquely) with a frame, i.e., with an injective linear ...
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Lagrangian invariant subspace of symplectic matrix

Suppose $S$ is symplectic matrix with only real eigenvalues. I need to prove that $S$ has Lagrangian invariant subspace, i.e. there is $L$ - Lagrangian, such that $S(L) \subset L$. I know that ...
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Do coordinate changes only affect antisymmetric matrices linearly?

Let there be an antisymmetric tensor field $\Omega_{ab}(q)$ where $q^i$ are coordinates on a 2N dimensional manifold. For context, this is a general symplectic form on phase space. I want to find a ...
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Is a hamiltonian isotopy real-valued of manifold-valued?

I have looked everywhere and I can't find a clear definition of hamiltonian isotopy. I have these definitions in my lecture notes but they are rather confusing Acording to the last definition a ...
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Prove that the Lie bracket of two symplectic vector fields is a Hamiltonian vector field

I am working on: Prove that the Lie bracket of two symplectic vector fields is a Hamiltonian vector field. More precisely, show that for all $X, Y ∈ \scr X(M, \omega)$ we have $i_{[X,Y ]}\omega = −dH$ ...
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Values of differential 2-forms on $k$-dimensional planes

I've found this demonstration (for this problem taken from "Mathematical methods of Classical Mechanics" by V. I. Arnol'd), and I could not decode this particular step: A $k$-dimensional ...
Lo Scrondo's user avatar
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Why is this identity of hamiltonian vector fields true: $ X_{F^t}+X_{{G^t}\circ (\varphi^t_F)^{-1}}=X_{F^t +G^t\circ (\varphi^t_F)^{-1}}$?

In the following proof, (source https://jmp.sh/X5KwZ1Zt) Using statement (i) as stated I only get $X_{F^t}+D\varphi_F^tX_{G^t}\circ (\varphi^t_F)^{-1}=X_{F^t}+X_{{G^t}\circ (\varphi^t_F)^{-1}}$ but ...
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The set $Ham(M,\omega)$ of Hamiltonian diffeomorphisms is a subgroup of the set of symplectic diffeomorphisms $Symp(M\omega)$

The set $Ham(M,\omega)$ of Hamiltonian diffeomorphisms is a subgroup of the set of symplectic diffeomorphisms $Symp(M\omega)$ is proven in the following lectures notes https://jumpshare.com/s/...
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A vector field is symplectic iff $\scr{L}$ $_{X}\omega=0$

I am trying to prove that vector field $X$ is symplectic iff $\scr{L}$$_{X}\omega=0$ $(M,\omega)$ symplectic manifold (compact, smooth and connected if this is necessary) If X is symplectic, then by ...
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4 votes
1 answer
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$Ham(M, \omega)$ acts transitively on $(M,\omega)$

Let $M$ be a compact and connected smooth manifold with a symplectic form $\omega$. $Ham(M, \omega)$ denotes the space of hamiltonian symplectomorphisms of $(M,\omega)$. I have the following ...
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Prove that the isotopy generated by a time-dependent symplectic vector field is a symplectomorphism

Let $M$ a compact and connected smooth manifold. Suppose $X_t$ is a time-dependent symplectic vector field and let $\phi_t$ be the isotopy generated by $X_t$. Prove that $\phi_t ∈ Symp(M, \omega)$ for ...
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