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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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 ...
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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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1answer
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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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39 views

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 ...
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54 views

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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38 views

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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1answer
46 views

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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1answer
79 views

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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1answer
28 views

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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38 views

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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35 views

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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1answer
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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

This is an extract from Duistermaat's Fourier integral operators. I'm having a hard time understanding the proof. My questions are three: How do I use the implicit function theorem to ...
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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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1answer
156 views

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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57 views

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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1answer
94 views

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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33 views

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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Why is Chirikov Map is a Twist Map?

Chirikov Map, also know as Standard Map, is define by $$ X(x,y)= x + y - \frac{k}{2\pi}\sin{2\pi x} \\ Y(x,y)=y - \frac{k}{2\pi}\sin{2\pi x}$$ the definition 4.1 given by Christophe Golé, in the book ...
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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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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 ...
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1answer
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Symplectic geometry spectrum

I have a question about a step in the proof of the following theorem from symplectic geometry: The theorem is: Given any ellipsoid $E:=\{ w \in \mathbb{R}^{2n}; \sum_{i,j =1}^{2n} a_{ij}w_iw_j \le ...
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Hamiltonian vector field and symplectic geometry

I want to show the following theorem: For any Hamilton function $H : M \rightarrow \mathbb{R}$ on some symplectic manifold $M$ and symplectomorphism $f : M \rightarrow M$ we have $X_{H \circ f} = ...
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Stokes' theorem and symplectic geometry

Let $V = \mathbb{R}^2,$ as a vector space then the Poincaré invariant is an integral $\int_{\gamma} \theta$ where $\theta = p dx $ is the symplectic 1-form and $\gamma$ a closed curve. Now, it is ...
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Symplectic form on a Hilbert Space is Closed

Let $\mathcal{H}$ be a Hilbert space over $\mathbb{C}$. Define a new vector space, $V$, over $\mathbb{R}$, which has, on the level of sets, $V = \mathcal{H}$ and for scalar multiplication (only with ...
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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 ...
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Definition of symplectic map

I just started reading on symplectic integrator of Hamiltonian system for my physics project, I don't quite understand some of the basic definitions here(see ...
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1answer
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Describe the space of solutions to a simple matrix equation

I would like to describe the space of solutions to the following matrix equation. Here $U$ and $V$ are two unknown real matrices with $n$ lines and $p$ columns (i.e. they belong to ...
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A variational approach to symplectomorphisms

Let $(\Sigma,\omega)$ be a compact symplectic surface, and let $\mathcal{D}$ be the group of diffeomorphisms of $\Sigma$. Is there some known variational approach to determine if an element ...
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1answer
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A function on a space of symplectic forms

Let $M$ be a smooth manifold and $\text{symp}(M)$ be the set of all symplectic forms on $M$. Let $\text{Diff}_{ 0}(M)$ be a connected component of difeomorphisms of $M$. Then, is there an explicit ...
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Differentiating an endomorphism

Let $(M,\rho)$ be a symplectic $2$-dimensional manifold, and let $J$ be a compatible complex structure on $M$, i.e. the symmetric $(0,2)$-tensor $$g(*,*) = \rho(*,J*)$$ is a Riemannian metric. Denote ...
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Wedge product equality of 2n-forms in 2n+2 dimension

I have a question that seems to me clear but i couldn't prove it. I have a diffeomorphism between the cotangent bundle of the n-sphere and the complex quadric as $$T^*(S^n)\to (S^n)^{\mathbb{C}},\ ...
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Tangent space of the space of compatible complex structures

Let $(M,\omega)$ be a symplectic manifold, and let $\mathcal{J}(M,\omega)$ denote the space of complex structures on $M$ which are compatible with $\omega$. I have been told the following fact: We ...
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Symplectic Geometry of 2-sphere in stereographic projection

I am trying to put the symplectic form of the 2-sphere defined by $\omega_u(v,w) := \langle u,v\times w\rangle,$ where $u \in \mathbb{S}^2$ and $v, w \in T_u\mathbb{S}^2$ in stereographic coordinates ...
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1answer
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Are the orbits of a symplectic toric manifold the fibers of its moment map?

A symplectic toric manifold, by definition, carries an effective torus action generated by a moment map. The orbits of the torus action are of course contained in the fibers of the moment map, but ...
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Symplectic reversing diffeomorphisms

Let $(M,\omega)$ be a compact symplectic manifold. Is there a diffeomorphism $f$ on M with $f^{*}\omega =-\omega$?
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Convex boundary of a symplectic manifold

Given a symplectic manifold $(M,\omega)$, suppose that $\partial M$ is of contact type. A Liouville field on a symplectic manifold is a vector field $X$ such that $\mathcal L_X \omega = \omega$. We ...
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1answer
51 views

symplectic change of variables

I must prove that the change of variables $\Psi: (p,q)\rightarrow(r,\alpha)$ such that $$q=\sqrt{2r}\cos(\alpha), \qquad p=\sqrt{2r}\sin(\alpha)$$ is symplectic. What I know is: $$\dot{q}=p, \qquad ...