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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Coordinate-free proof of the hamiltonian character of the geodesic flow

Let be $(M,g)$ a pseudoriemannian manifold. Let us identify the tangent and the cotangent bundles through the musical isomorphism $g^\flat:u\in TM\to g(u,\cdot)\in T^\ast M.$ It is well known ...
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symplectic lie algebra is simple

The symplectic lie algebra defined by $sp\left(n\right)=\left\{ X\in gl_{2n}\,|\, X^{t}J+JX=0\right\}$ when $J=\begin{pmatrix}0 & I\\ -I & 0\end{pmatrix}$. So $X\in sp\left(n\right)$ is of ...
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Symplectic basis $(A_i,B_i)$ such that $S= $ span$(A_1,…,A_k)$ for some $k$ when $S$ is isotropic

Let $(V,\omega)$ be a symplectic vector space of dimension $2n$. How can I show that for an isotropic subspace $S \subset V$, there exists a symplectic basis $(A_i,B_i)$ for $V$ such that $S= $ ...
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Symplectic basis $(A_i,B_i)$ such that $S= $ span$(A_1,B_1,…,A_k,B_k)$ for some $k$ when $S$ is symplectic

Let $(V,\omega)$ be a symplectic vector space of dimension $2n$. How can I show that for a symplectic subspace $S \subset V$, there exists a symplectic basis $(A_i,B_i)$ such that $S= $ ...
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basis free volume form for a symplectic vector space

It's easy to show, using a symplectic basis, that if $\omega$ is a symplectic form on a $2n$-dimensional vector space $V$, then $\omega^n \neq 0$. I'd like to be able to prove it without choosing a ...
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reference for strongly continuous semi-groups

At the moment I am trying to understand the proof of the Fredholm property in Salamon's notes on Floer homology. There I came across the notion of an unbounded operator on a (real) Hilbert space which ...
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301 views

Harmonic Oscillator and Quadrature

Consider the simple harmonic oscillator $\frac{d^2p}{dt^2}=-p$ as a Hamiltonian system with Hamiltonian given by $H=\frac{1}{2}p^2+\frac{1}{2}q^2$. The famous Liouville theorem for integrable systems ...
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About $\operatorname{Spin}^\mathbb{C}$ structures

Let $X$ be a orientable 4-manifolds. I know that $X$ can be endowed with $\operatorname{Spin}^\mathbb{C}$ structures by the choice of integral lift of $w_2(X)$ (second Stiefel-Whitney class of ...
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Is the Poisson tensor associated to a left invariant symplectic form, also left invariant?

Given a left invariant symplectic form $\omega$ on a Lie group $G$, the Poisson tensor associated to $\omega$ is given by $$\pi(df,dg)=\omega(X_f,X_g)$$ where $X_f$ is the hamiltonian vector field ...
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Tangent space of Cotangent bundle at zero section?

Let $M$ be a differentiable manifold with cotangent bundle $T^*M$. How can I prove that $T_{(p,0)}T^*M$ is naturally isomorphic to $T_pM\oplus T_pM^*$? If this true, then I think I could prove that ...
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How to prove that a certain action is hamiltonian?

Reading a paper I had the need to complete a proof, and come up with a certain argument(see below). My question is: could I reduce it to a special case of some theorem? I ask this question in order to ...
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is any hamiltonian system with just one degree of freedom completely integrable?

An hamiltonian system with $n$ degree of freedom is said to be completely integrable when there exists an system $f_1,\ldots,f_n$ of first integrals mutually Poisson-commuting, such that ...
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Floer theory or Floer homology, an introduction for physicists needed

I need an introduction to Floer theory that's suitable for perhaps a beginning math grad student or a 2nd year physics grad student. The wiki article is sufficiently over my head that it reads as ...
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compact symplectic manifolds

Why there is no compact symplectic submanifold with dimension greater than 2 in $\mathbb{R}^{2n}$ ?
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504 views

Symplectic form on a complex manifold

I am a little muddled and am hoping I can get some clarification about forms in a complex manifold. Since I am only concerned with local issues, consider $M = \mathbb C^n$ as a complex manifold. So ...
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References for Smale's Geometry

I read that Smale recast classical mechanics in terms of symplectic geometry. I know a bit about classical mechanics but nothing about symplectic geometry. Are there any writings from Smale on this ...
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Why does symplectic geometry have many applications in mathematics

It is not quite intuitive , at least from its origin. Could any one can give me an intuitive explanation?Thank you!