Mathematics Stack Exchange
Mathematics Stack Exchange

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.

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. Symplectic geometry has its origins in the Hamiltonian formulation of classical mechanics where the phase space of certain classical systems takes on the structure of a symplectic manifold. For more information please consult the Wikipedia page on symplectic geometry.

1282 questions
1
vote
0 answers

Symplectic nonequivalent subspaces of a given dimension in a symplectic space

Let us be given a symplectic space of dimension $2n$ $(V, w)$. Symplectic equivalent subspaces $L_1, L_2$ are such subspaces $V$ for which there exists a symplectic operator $A$ such that $L_1=AL_2$. The question is as follows: how many symplectic…
AndrewGap
  • 101
1
vote
0 answers

The map not induced by hamiltonian flow.

Some time ago I needed to solve the following problem: We have a torus with $\mathbb{T}^2$ with coordinates $(x \ mod \ 1, y \ mod \ 1)$ and symplectic form $\omega = dx \wedge dy$. Given function $F : \mathbb{T}^2 \to \mathbb{T}^2$ $F(x,y) = (x + …
Turtle5
  • 357
1
vote
0 answers

conic Lagrangian submanifold with boundary

I'm reading through the paper "Lagrangian Intersection and the Cauchy Problem" by Melrose and Uhlmann, and I'm having trouble with the definition of intersecting pair of Lagrangian manifolds. I am aware of the definition of conic Lagrangian…
Kerwin Yi
  • 75
1
vote
0 answers

Weinstein theorem and symplectic structures on associate bundles

The following is a theorem cited as a result of Weinstein in McDuff & Salamon's Introduction to Symplectic Topology, Section 6.2. Theorem: let $G \rightarrow Symp(F,\sigma)$ be a Hamiltonian action. Then every connection on a principal $G$-bundle…
topolosaurus
  • 1,943
1
vote
0 answers

Negative intersection of symplectically embedded planes in $\Bbb R^4$

Consider $\Bbb R^4$ with standard symplectic form $dx_1\wedge dy_1+dx_2\wedge dy_2$. This determines an orientation of $\Bbb R^4$ so that $(\partial_{x_1},\partial_{y_1},\partial_{x_2},\partial_{y_2})$ is a positive basis, and we can define…
user302934
  • 1,698
1
vote
0 answers

Symplectic geometry prerequisites

In looking at various notes on Symplectic Geometry (SG) as well as the book by McDuff/Salamon, it is either said or implied that the interested student should have seen the Hamiltonian before. It goes without saying that smooth manifold theory is…
NoodleNami
  • 163
1
vote
1 answer

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 path $\gamma(t)=(q(t),p(t))$ such that $\dot…
Someone
  • 4,737
1
vote
1 answer

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({$\psi_{t}$}) = [$i(X) \omega$] (here $\omega$ is the…
Steppewolf
  • 184
1
vote
0 answers

Symplectomorphism between square and disk of the same area

I understand that the disk and square of area $1$ are symplectomorphic in $\mathbb{R}^2$ but I can't construct a map that preserves the standard form on $\mathbb{R}^2$. How is this done?
mtheorylord
  • 4,274
1
vote
0 answers

Why $\mathbb{S}^0$ has a symplectic structure?

There's an excercise problem in Heckman's symplectic geometry book that says $\mathbb{S}^n$ is a symplectic manifold iff $n=0$ or $n=2$, but I don't see the structure over $\mathbb{S}^0$.
Gyadso
  • 219
1
vote
1 answer

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?
user220607
  • 19
1
vote
0 answers

Duistermaat Heckmann formula

Which physical concepts are related to the Duistermaat-Heckmann formula, and are symplectic schur functions related to it by any chance?
AkatsukiMaliki
  • 968
1
vote
1 answer

Definition of bivector field on a manifold.

I am reading this article: http://arxiv.org/abs/1112.5037v1 . In this, it defines a symplectic manifold as a manifold equiped with a nondegenerate bivector field $\pi$ that is Poisson. l want to understand what is a bivector field on a manifold. I…
Yamid Yela
  • 48
1
vote
1 answer

Isomorphism of tangent and cotangent spaces induced by a symplectic structure on a manifold

If $(M^{2n}, \omega^2)$ is a symplectic manifold, then for each tangent vector $v \in T_xM$ for some $x \in M$ we may associate a $1$-form $\omega_v^1 \in T_x^* M$ by $\omega_v^1(u) = \omega^2(u,v)$. The map carrying a vector to it's associated…
BroDoYouEvenCalculus
  • 67
1
vote
1 answer

Is the Cartesian product of Lagrangian submanifolds a Lagrangian submanifold?

In my research, I have found a statement where a submanifold of $\mathbb{C}^N$, which is a straight line in each plane $\mathbb{C}$ is a Lagrangian submanifold. Similarly it seems that a submanifold of $\mathbb{C}^N$ which is a circle in each plane…
Mtheorist
  • 333
1 2 3
4
5 6 7