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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What's the definition of weight in localization theorem?
I am currently reading a book on symplectic topology. I may have skipped some pages so find it confusing about the Duistermaat-Heckman theorem. In the book it states that
Assume Hamiltonian function $...
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Liouville measure in the energy level
Again banged my head trying to make sense of the Mechanics course and hit a rock bottom with another problem. I would really appreciate the step by step explanation for the results of it.
We have a ...
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Canonical transformations on cotangent bundle
We got as part of homework for Mechanics this exercise. As the course was a little chaotic I barely got grip of some notions and I feel a bit lost so any solution would be welcomed.
Let V be a ...
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Why is symplectic geometry the natural setting for classical mechanics?
I was reading this very nice document, to understand why symplectic geometry is the natural setting for classical mechanics. I more or less understood why there is naturally a 2-form that arises. ...
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A query about Atiyah's proof of the convexity of moment map
This is about proving connectedness of level sets of moment map of a $\mathbb T^n$ $\implies$ convexity of image of moment map for $\mathbb T^{n+1}$ action. I am following Ana Cannas's wonderful ...
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On the proof of localization in symplectic geometry
I was working on the proof of Duistermaat-Heckman theorem in Introduction to Symplectic Topology by Dusa McDuff. He used a lemma called localization. It can be found on page 194. You can find
the ...
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25 views
Symplectic transformation of ellipsoid in standard symplectic space
Given the 4-dimensional standard symplectic space $\mathbb{R}^4$ with Darboux-Coordinates $(z_1,z_2,z_3,z_4)=(x_1,x_2,y_1,y_2)$ and the symplectic structure $\omega=\sum dx_j \wedge dy_j$, i am trying ...
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Symplectic quotient and Lagrangian subspace of an infinite dimensional symplectic vector space
Suppose that $(W,\Omega)$ is a symplectic vector space, $L \subset W$ is a Lagrangian subspace and $G \subset W$ is a coisotropic subspace ($G^\perp \subset G$) such that $L \cap G^\perp = \{0\}$. (...
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Connected sum of Lagrangians also a Lagrangian?
Given a symplectic manifold $(M,\omega)$ and pair of compact Lagrangians submanifolds $L_1,L_2$ inside it, do we know that there is a Lagrangian submanifold in M homeomorphic to their connected sum $...
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Is the group of symplectomorphisms normal?
Let $(M, \omega)$ be a symplectic manifold.
Question: Is the group of symplectomorphisms a normal subgroup of the group of all diffeomorphisms?
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Symplectic or geometric integrators for port-controlled Hamiltonian
Similar to this question, I'm interested in finding a symplectic integrator for a specific generalized (port-controlled) Hamiltonian problem.
The problem is as follows:
\begin{align}
\begin{bmatrix}
{...
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Examples of symplectromorphism other than $Sp(V)$
A linear symplectic form $ \omega$ on a vector space $V$ induces a symplectic structure (also denoted $ \omega$) on $V$ via the canonical identification of $TV$ with $V \times V$.
The symplectic ...
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Energy bounds for pseudoholomorphic curves without symplectic structure
Let $(M,J)$ be a compact almost complex manifold and $C>0$ a constant.
Gromov's compactness theorem states that the space of $J$-holomorphic curves $u:S^2\to M$ with energy bounded above by $C$ ...
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Unitary group acts transitively on the Lagrangian Grassmannian
I am trying to prove that the unitary group associated to an $\omega$-compatible complex structure $J$ acts transitively on the Lagrangian Grassmannian $\mathcal{L}(V)$.
I know that for a symplectic ...
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Volume of certain subsets of $T^*{S^{3}}$
Consider $S^{3}$ equipped with the round metric of unit radius. The metric naturally gives a metric on the cotangent bundle, so we can consider the subset $$S_{t} = \{v \in T^{*}S^{3} : |v|^{2} < t\...
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Quantization on Lagrangian submanifold
I am reading a paper by Karasev and quite confused with the abstract definition of quantization.
To begin with, we define a map from the phase space $\mathbb{R}^n_x\times\mathbb{R}^n_p$ to $\mathbb{C}...
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1answer
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Pullback of Fubini-Study form on $\mathbb {CP}^1$
Question: Let $\varphi_S: S^2\setminus \{N\} \to \mathbb C$ be given by $\varphi_S (x_1, x_2, x_3) = \left(\frac{x_1}{1 - x_3}, \frac{x_2}{1 - x_3}\right)$, i.e., the stereographic projection. Also ...
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Hamiltonian vector field proof?
Given a manifold $M$ and a symplectic form $\omega \in \Omega^2(M)$, let $j:Y\to M$ be a symplectic submanifold (i.e, $j^*\omega \in \Omega^2(Y)$). Now, let $H \in C^{\infty}(M)$ be a smooth function ...
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1answer
70 views
Example of two homotopically equivalent manifolds such that one admits a symplectic structure and the other does not
A smooth manifold $M$ admits a symplectic structure if there is an alternating
non degenerate $2$-form $\omega \in \Lambda^2(M)$ that is also closed i.e. $d\omega = 0.$
Usually we can express ...
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3answers
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Interpreting the closed condition of a symplectic form
In McDuff's What is Symplectic Geometry?, she writes: "the fact that $\omega$ is closed means that the symplectic area of a surface $S$ with boundary does not change as $S$ moves, provided that the ...
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Symplectic and Euclidean structure invariance
Consider a $2n$ real symplectic space - the usual $\mathbb R^{2n}$.
Suppose that the same space could be endowed too with an Euclidean structure, by which the vectors of the symplectic basis are ...
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1answer
60 views
Finding an explicit formula for a Hamiltonian vector field
I've been looking at this question: Existence of vector field given a smooth function
That is: Given a symplectic manifold $M$ of dimension $2n$, with a symplectic form $\omega \in \Omega^2(M)$, do ...
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1answer
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Where is it used that the symmetry conserves the symplectic form in noether's theorem.
For the proof of Noether's theorem, it seems like that the only thing that's important for the symmetry map $S_g : M \to M$ is that it conserves the Hamiltonian (which will then imply that the moment ...
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Unit of (Lagrangian) Floer cohomology
Let $(M,\omega)$ be a symplectic manifold and $L \subseteq M$ be a compact Lagrangian in M. My question is what is a geometric/natural representative for the unit of the Floer cohomology $HF(L,L)$? Or ...
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Composed symplectic integrator for non-separable hamiltonian
Firstly, I have seen Symplectic integration for non-separable hamiltonian, and I do not believe it answers my question.
I have been attempting to modify a symplectic integrator that I wrote a while ...
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Connecting the Fibres of a Lefschetz Fibration With Disconnected Fibres.
The following is an exercise set by Chris Wendl in his book Holomorphic Curves in Low Dimensions. I'm fairly new to the subject, and wasn't sure how to approach it, so any help with it would be ...
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Biconformal space and curvature
I've found very few contributions about the so called Biconformal Space, "a curved phase space".
I was sure that in general phase spaces are cotangent bundles naturally equipped with a symplectic 2-...
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29 views
classical mechanics in coadjoint orbits
We know that coadjoint orbits are symplectic manifolds, and they can be used to find unitary representations of lie groups and stuff, and it's also related to quantization. However, is it true that ...
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0answers
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1-form sufficiently $C^1$-close to the zero section
Question: If $\mu$ is a 1-form sufficiently $C^1$-close to the zero 1-form, then
$$\{(p, \mu_p) ; p \in M, \mu_p \in T_p^*M \}\cong\text{Graph}\ f$$
for some diffeomorphism $f: M \to M$.
...
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1answer
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Euler characteristic expression in terms the number of fixed points of an $\mathbb{S}^1$ action
I have found in a paper* that I am reading that
Given $(M,J)$ compact (smooth) manifold with an almost complex structure $J$, if we have an $\mathbb{S}^1$ action with isolated fixed points then $ \...
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Generalized Poincaré Lemma
I'm reading the proof of an improved version of Poincaré's Lemma on Ana Cannas da Silva's Lectures on Symplectic Geometry, page 40. I am terribly confused. Here's the setup: $U_0$ is a tubular ...
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2answers
173 views
The physical meaning of a symplectic form.
So I've studied a bit about symplectic geometry, and I know that phase space is a symplectic manifold, and the symplectic form induces a poisson bracket. However, what is the physical meaning of the ...
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Inverting the WZW Lagrangian to its Poisson bracket
Could someone give me a hint or how one should start for inverting the WZW Lagrangian or equivalently the Kirillov-Kostant 2-form:
$$\Omega = \frac{K}{4\pi} \int_0^{2\pi} \mathrm d x \text{ Tr} (g^{-1}...
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Chern class of holomorphic symplectic manifold
Let $M$ a complex surface and $\omega\in H^0(\Omega_M^2,M)$ a non degenerate holomomorphic form.
I've read somewhere (without proof), that then the first chern class of the symplecitc manifold $(M,Re~ ...
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Negative Tight Contact Structures on $S^3$
A famous theorem of Eliashberg says that given a positive tight contact form $\lambda$ in $S^3$ there is a diffeomorphism $\Phi: S^3 \to S^3$ such that
$$
\Phi^*\lambda = f\lambda_0,
$$
where $f: S^3 \...
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1answer
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Chern classes of symplectic manifolds
I have seen that people assign chern classes to the tangent bundle of symplectic manifolds. This confuses me, because to my knowledge chern classes detect differences in the complex structures of ...
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Naive deformation quantisation of a symplectic manifold
Consider a $2n$-dimensional symplectic manifold $(M,\omega)$ with an open cover $(U_i)$ such that $U_i$ is symplectomorphic to $T^{*}V_i$ of some non-symplectic manifold $V_i$ for all $i$. Since each ...
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1answer
37 views
how to Prove that the set of fixed points of a Hamiltonian action of a torus on a symplectic manifold is a symplectic submanifold.
who can explain how to Prove that the set of fixed points of a Hamiltonian action of a torus on a symplectic manifold is a symplectic submanifold?
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Duistermaat Heckmann formula
Which physical concepts are related to the Duistermaat-Heckmann formula, and are symplectic schur functions related to it by any chance?
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Integrable system vs lagrangian fibration
Every complete integrable system $I:M \rightarrow \mathbb{R}^n$ is a regular langrangian fibration on a dense subset of the symplectic manifold $M$.
It is also known that locally every lagrangian ...
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1answer
90 views
Euler field for magnectic symplectic structure in $T^*Q$
Let $Q$ be a differentiable manifold, $\pi\colon T^*Q\to Q$ denote its cotangent bundle, and $B \in \Omega^2(Q)$ be a closed form. I'm playing around with some things, and I'm not sure whether it ...
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1answer
43 views
Why does $g\circ \exp(tv) \circ g^{-1}$ give the one-parameter group of diffeomorphisms generated by $g_* v$?
I have a question regarding the following proof from Cannas da Silva - Introduction to Symplectic and Hamiltonian Geometry:
Let $(M, \omega)$ be a symplectic manifold, and let $\alpha$ be a 1-form ...
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1answer
31 views
Why the Hamiltonian is constant along the integral curves of the hamiltonian vector field?
Let $H$ the Hamiltonian of a system and $\gamma $ an integral curve of the Hamiltonian vector field, i.e. if $\gamma (t)=(q(t),p(t))$ and $H(p,q)$ is the Hamiltonian, then $$\begin{cases} \dot p=-H_q\\...
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1answer
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Intuition about Poisson bracket
I've been reading about Hamiltonian mechanics which in its mathematical description uses Poisson manifolds
From my limited understanding, on a Poisson manifold $M$ we can look at the Poisson bracket ...
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1answer
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Constructing diffeomorphisms of moduli spaces of $J$-holomorphic curves
Let $M^{2n}$ be a smooth manifold admitting two almost complex structures $J_0$ and $J_1$.
Suppose that $J_0$ and $J_1$ are both regular in the sense that the moduli space $$
\mathcal{M}_i:=\mathcal{...
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2answers
47 views
Identity Involving Lie Derivative and Local Flows
I'm trying to show,
$$ \frac{d}{dt} \varphi_t^* \omega = \varphi_t^* \left( \mathcal{L}_{X_t} \omega \right)$$
but I have another question as well. Every case in which the lie derivative is ...
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1answer
44 views
introduction to hamiltonian actions and moment maps.
I'm trying to find some good articles or books to learn about moment maps and hamiltonian actions. I ddo have some basic differential geometry (and representation theory) knowledge, but not really ...
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What generalizes the definition of a symplectic manifold?
Reference required for the following:
A vector field $v$ on a symplectic manifold $(M,\omega)$ is that preserves the symplectic form $\omega$ along its direction
$$\mathcal L_v \omega =0 \,.$$
In ...
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1answer
48 views
Rescaling a symplectic form and integral cohomology
Let $(M,\omega)$ be a symplectic manifold. I am trying to understand a procedure which seems so obvious that its implications are omitted in any article I could read. I encountered the following: ...
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1answer
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Deforming antiholomorphic involutions
Let $(M,J)$ be a compact smooth almost complex manifold. We can "deform" $J$ as follows: if$A$ is a smooth section of the endomorphism bundle $\mathrm{End}(TM)\to M$ satisfying
$
AJ=-JA,
$
it follows ...
