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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Existence of specific non-symplectic manifolds
I am fairly new to symplectic geometry but I started wondering about this :
From the definition of a symplectic manifold we can extract that $M$ has to be even dimensional , orientable , and in the ...
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Correspondence between hamiltonian flows
Suppose I have an hamiltonian function $H$ that is fiberwise homogenenous of degree 2 , i.e, $H(q,sp)=s^2H(q,p)$,in $(T^*M,\omega)$ the cotangent manifold with the canonical symplectic structure and a ...
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Value of a hamiltonian path along a function closely related to the hamiltonian function
Consider $T^*M$ with the canonical symplectic structure.
Let $H:T^*M\rightarrow \mathbb{R}$ be an hamiltonian function and $h:\mathbb{R}\rightarrow \mathbb{R}$ a smooth function. Let $\gamma(t)$ be ...
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$(V,\omega)$ with $W,W'$ isotropic, then there exists $\phi \in Aut(V)$ such that $\phi(W)=W'$ iff $\dim(W)=\dim(W')$
Let $(V,\omega)$ be a symplectic vector space. Let $W,W'$ be two isotropic subspaces. I want to see that there is symplectomorphism $\phi:V\rightarrow V$ such that $\phi(W)=W'$ if and only if $\dim(W)=...
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Every isotropic subspace is contained in a lagrangian subspace.
Let $(V,\Omega)$ be a symplectic vector space. A subspace $L$ is called Lagrangian if $L^\perp=L$. A subspace $W$ is called isotropic if $W\subset W^\perp$.
For any isotropic subspace $W$ there is a ...
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Can a lagrangian submanifold of a 4-dimensional real space be diffeomorphic to sphere or real projective plane?
Is it possible to find a lagrangian submanifold of $\mathbb{R}^4$, which is diffeomorphic to $S^2$ or $\mathbb{R}P^2$?
As I have found, the answer is no for $S^2$, because $S^2$ is simply connected, ...
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How many Lagrangian subspaces are there over $\mathbb{F}_2$?
Let $(V,\omega)$ be a finite-dimensional ($\dim(V)=2n)$ symplectic vector space over $\mathbb{F}_q$. How many Lagrangian subspaces of $V$ are there? I am particularly interested in the case $q=2$.
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The Heisenberg group/algebra and Symplectic Vector Spaces
I have some questions about the relationship between Heisenberg groups/algebras and symplectic vector spaces. This is my first time properly dealing with many of these topics, so please be patient if ...
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Computing a differential on the loop space of a Symplectic manifold
Suppose we have a symplectic manifold $(M,\omega)$ such that $\omega$ is exact ,i.e, $\omega=d\alpha$. Now let $L_0$ and $L_1$ be two lagrangian submanifolds and consider $\Omega(L_0,L_1)$ to be the ...
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Show that $S(q^i,s^j,t)=\frac{1}{2t}||\mathbf{q}-\mathbf{s}||^2$ generates canonical at $t=0$
I'm currently working through Introduction to Mechanics and Symmetry, specifically the section on Generating functions of canonical transformations. I am having an issue with the following problem:
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Reality check on derived closed forms
Apparently, there's a bicomplex characterisation of derived closed forms with on Joyce's talk slides and notes. Is the $[i]$ here meaning shifting up in cohomological degreeing?
In other words, is it ...
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Connection 1-form, symplectic potential?
I recently encountered a formula that fell a little bit from the sky:
Given: A symplectic manifold $(M,\omega)$, a Hermitian line bundle $\pi:B\rightarrow M$, a connection $\nabla$ on $B$ and a ...
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1answer
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Identity for projection and inclusion of the canonical one-form on the cotangent bundle
I'm currently reading through Introduction to Mechanics and Symmetry by Marsden and Ratiu, specifically the section on Cotangent bundles. I'm trying to do the following exercise:
Let $N$ be a ...
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How to recover a symplectomorphism from a generating function of type V?
In McDuff and Salamon's Introduction to symplectic topology, they define generating functions of type V for a symplectomorphism in lemma 9.2.1.
Lemma 9.2.1.
If $\psi$ satisfies $\|{d\psi- Id1}\|\leq \...
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Generating functions for the iterates of a symplectomorphism
$\newcommand{\coloneqq}{\colon\!=}$
In an exercise at the end of section "5.1 Periodic points" of Ana Cannas da Silva's book "Lectures on Symplectic Geometry" it is stated that, ...
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Trouble understanding notation of symplectic forms, differential forms and differentials
I found the exerpt below in the book "Introduction to Symplectic Topology".
I understand that a 2-form $\omega(p,\zeta, \zeta')$ on $\mathbb R^{2n}$ is a function that maps $\mathbb R^{2n}\...
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What is meant by *linear* symplectomorphisms
From my modest knowledge of differential geometry, the Symplectomorphism group, $Sympl(M, \omega)$, on a symplectic manifold, $(M, \omega)$, is a subgroup of the diffeomorphism group that satisfies $s^...
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Must the action of a torus on a manifold have a fixed point if each circle subaction has a fixed point?
Let $T = (S^1)^k$ act smoothly on a compact manifold $M$ via $\rho : T \times M \to M$.
Suppose for every cocharacter/group map $\sigma : S^1 \to T$, the circle subaction $\rho \circ \sigma : S^1 \...
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1answer
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Restriction of symplectic form on submanifold
Consider a symplectic manifold $(M,\omega)$, for a 2 dimensional submanifold $N \subset M$, do we always have $\omega|_{N}$ gives a volume form on $N$?If not, is there any condition we can put on $N$ ...
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Maximum number of independent constants of motion in involution
Consider a holonomic system. Let $M$ be the phase space $T^{\ast}Q$ where $Q$ is the configuration space with $\dim(Q)=n$. Two functions $f,g: M \to \mathbb{R}$ are in involution if they Poisson-...
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Finding the canonical coordinates on the sphere using an area form
I'm currently reading through Introduction to Mechanics and Symmetry to learn some symplectic geometry and I am encountering a problem. Question 5.1-1 says " Show how to construct (explicitly) ...
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Symplectic Hodge Star and Koszul differential
Let $M = \text{Spec}(R)$ be a symplectic affine variety of dimension $n$ with the symplectic form $\omega$. There is the symplectic Hodge star $\star: \Omega^k_{M} \to \Omega^{2n-k}_{M}$ given by the ...
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Symplectic potential determines local trivialization?
Perhaps this is pretty basic, but somehow I'm not getting it:
Given a symplectic manifold $(M,\omega)$, a Hermitian line bundle $B \rightarrow M$, pick a symplectic frame and set
$$ \theta = \frac{1}{...
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Metric determined by complex structure and volume form
Consider a Riemann surface that carries a complex structure and a volume form, can we then determine a metric by these two condition?
I know the answer is true when the complex strucutre and volume ...
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More Examples of Positive Measures on Manifolds
Given a smooth manifold $M$, there are several ways of constructing measures on $M$. The most common procedure I've seen is by starting with a $(0,2)$ tensor field $T$ on $M$, and defining for each ...
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Symplectisation of contact manfiold and Legendre submanifolds
If $P$ is a contact manifold with contact form $\alpha$ , the symplectisation of $P$ is the symplectic manifold $P \times \mathbb{R}$ with symplectic form $d(e^t \alpha)$. I'm struggling to show the ...
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Poisson structure identity
I'm attempting a question under Hamiltonian Dynamics. We are given that $\omega^{ab}$ is an antisymmetric matrix such that it's components depend on coordinates $x^a$ and such that the Poisson bracket
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Heisenberg group - representation - symplectic manifold
Given a symplectic vector space $(V,\omega)$ of dimension $2n$ ($V$ being a symplectic manifold), a smooth function $\psi \in C^\infty(M)$, a translation operator $T$ acting on $V$ via $(T(Y)\psi)(X) =...
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Time-dependent Hamiltonian isotopy and Hamiltonian symplectomorphism
I am having some difficulty understanding the definitions of Hamiltonian isotopy and Hamiltonian symplectomorphism. I know that if we have a Hamiltonian function $H: M \to \mathbb{R}$, we have a ...
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Geodesics in the symplectization of a contact manifold
As the title suggests, are there any reference that deals with geodesics in the symplectisation $$\Bbb R \times X$$ for a contact manifold $(X,\alpha)$. We equip the symplectisation with the ...
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1answer
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Hamiltonian diffeomorphisms generated by periodic Hamiltonian function
Let $\phi_{t}$ be a family of Hamiltonian diffeomorphisms on a symplectic manifold $(M,\omega)$, generated by a family of Hamiltonians $H(x,t)$. I'm trying to show that the family of Hamiltonians is ...
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Functions vanishing to infinite order “in $1$-mean”
When proving certain unique continuation theorems for classes of functions which satisfy some type of PDE inequality, people often talk about functions which vanish to infinite order at a point $x_0$ ...
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1answer
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Showing that $\mathcal{L}(n)\cong U(n)/O(n)$
I have been reading the book Introduction to Symplectic Topology, page $50$, and now we want to see that $\mathcal{L}(n)=\mathcal{L}(\mathbb{R}^{2n},\omega_0)\cong U(n)/O(n)$. To see this we have the ...
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Darboux theorem for the symplectisation of a contact manifold
I'm wondering if there is some "nice" version of Darboux's theorem that can be applied in the case of a symplectisation of a contact manifold $(X,\alpha)$ with the canonical symplectic form. ...
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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 ...
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1answer
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Flux Homomorphism has right inverse
I'm reading about the Flux homomorphism in Symplectic Topology and I'm trying to show that it is surjective.
I know that if $\psi_{t}$ is the flow of a symplectic vector field $X$, then Flux({$\psi_{t}...
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Noether‘s Theorem and Moment Maps
Noether‘s Theorem says that every continuous symmetry of a physical system (i.e., a Lie group action on phase space ${\bf R}^{2n}$ preserving a Hamiltonian $H$) leads to a conservation law (i.e., a ...
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Is this true lemma in the proof of momentum map $J$ is a Poisson map?
Suppose $(M, \omega)$ be a symplectic manifold equipped with $G$-action $\Phi : G \times M \to M$. Assume $\Phi^*_g \omega = \omega$.
Let $J$ be an equivariant momentum map of $M$, $J : M \to \mathrm{...
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1answer
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Darboux chart compatible with Lagrangian submanifold
Let $(M,\omega)$ be a symplectic manifold of dimension $2n$ and let $L\subset M$ be a Lagrangian submanifold, that is $\dim L=n$ and $$\omega_p(u,v) = 0$$ for all $u,v\in T_pL$ holds for every point $...
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Determining the dimension of the Lagrangian Grassmannian through action of symplectic group
After reading chapter 9 in Lee's book on smooth manifolds, I'm trying to figure out the dimension of the Lagrangian Grassmannian in $\mathbb{R}^{2n}$, and I'm wondering if anyone can help me out.
I ...
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Why $T_{\alpha}:T^*X\to T^*X$ is a diffeomorphism? With $\alpha$ not necessarily a closed form.
I wanna know why if $\alpha$ a 1-form on $X$, then the next map is a diffeomorphism.
$$T_{\alpha}:T^*X\to T^*X$$
$$\hspace{2.4cm}\beta \mapsto \beta + \alpha_{\pi(\beta)} $$
I know if $\alpha$ is a ...
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Coordinate expression for the Cotangent Lift
Hello dear math people,
I find myself having difficulties to get a concrete coordinate representation of the contangent lift of the configuration space of a mechnanical system. Setupwise, we have a ...
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Product Rule for Derivatives of Column Vectors
I've just started working through Salamon's Introduction to Symplectic Topology. On page 23, I'm struggling to understand some of his working.
He defines the Poisson bracket by $ \{ A, B \} = -\nabla ...
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Is composition of discrete Hamiltonian flows integrable?
Consider $\Bbb{R}^2$ with the usual symplectic form $$\omega = dx \wedge dy$$
For a function $H \colon \Bbb{R}^2 \to \Bbb{R}$, let $X_H$ be the Hamiltonian vector field. Then the map $\Bbb{R}^2 \to \...
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Curvature form on Riemann Surfaces
I am trying to understand the basic construction in P. Biran's paper "Lagrangian Barriers and Symplectic Embeddings". At the begining (2.1), there is a construction which relies on the ...
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Why $\frac{d}{dt}\phi_t^*\omega_t=\phi_t^*(\mathcal L_{v_t}\omega_t+\frac{d}{dt}\omega_t)$?
let $U\subset \mathbb R^n$ open and $\omega_t\in \Omega^2(U)$ a differential 2-form with a parameter $t\in \mathbb R$.
Let $v_t\in \mathcal X(U)$ a vector field and $\phi_t$ its flow.
I don't ...
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1answer
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Compactly supported hamiltonian diffeomorphisms
I'm having trouble with an Exercise (12.3.6) from McDuff's and Solomon's book, "Introduction to Symplectic Topology" (3rd Ed.).
The goal is to prove the monotonicity of the symplectic ...
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Differential Forms on a Symplectic Manifold
Let $M$ be a symplectic (algebraic) variety over a field $k$ of dimension $2n$ with a symplectic form $\omega$. Is the map $\Omega^{k}_{M} \to \Omega^{2n-k}_{M}$ given by a multiplication by $\omega^{...
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1answer
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Image of a coadjoint orbit of SU(3) under a moment map
Let M be a coadjoint orbit of dimension 6 of $SU(3)$, and let T be the maximal torus in $SU(3)$. If we denote $\mu : M \longrightarrow \mathbb{R}^2$ the moment map associated to the action of T on M, ...
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Vector Bundle definition compared to direct product
I read that the simplest vector bundle is $E = M \times V$ where $M$ is a manifold and $V$ is some $r$-dimensional vector space. I suppose my confusion is with notation. My understanding is that if I ...
