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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
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Symplectomorphisms preserves Hamiltonian equations

Let $(M_1,\omega_1)$, $(M_2,\omega_2)$ be symplectic manifolds and let $\psi:(M_1,\omega_1)\rightarrow (M_2,\omega_2)$ be a symplectomorphism. Consider a Hamiltonian $H\in\mathcal{C}^\infty(M_2)$. Show that a curve $t\mapsto \gamma(t)\in M_1$ solves…
Giulio Binosi
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existence of symplectomorphism

Let $U,V\subset\mathbb{R}^2$ be two bounded opens of same area, wrt to the standard symplectic form $\omega_0$. If there's a diffeomorphism $\phi:U\to V$ preserving orientation, show that there exists a symplectomorphim form $(U,\omega_0)$ to…
Doug
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Area of flux homomorphism in symplectic topology

Let $(M,\omega)$ be a symplectic manifold. Let $f:M \to \mathbb R$ be a smooth function. We have vector fields $X_f$ defined by $\omega(X_f,)=df$. Let $\phi_t$ be the flow of $X_f$ and let $\gamma: \left[a,b\right] \to M$ be a smooth curve. We…
fidy
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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 $$\begin{cases} \dot p=-H_q\\ \dot q= H_p\end{cases}.$$ In wikipedia the say…
user623855
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Symplectic structure on a covering manifold

How to show that a covering manifold of a symplectic manifold admits a symplectic structure? More precisely, let M be a $2n$-manifold and $(N, \omega)$ be a symplectic 2n-manifold. If there exists a covering $\pi: M\to N$, then $\pi^*\omega$ is a…
usr1988
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Construct a nontrivial symplectomorphism of cotangent bundle

I have tried to prove that exercise 4 on the page 20, Lectures on Symplectic Geometry, Ana Cannas da Silva, which is available on professor's website: https://people.math.ethz.ch/~acannas/Papers/lsg.pdf Let $X$ be an arbitrary $n$-manifold, and let…
user530422
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Momentum Map-Submersion

Let $(M,\omega)$ be a symplectic manifold and $G$ a Lie group acting hamiltonian on $M$, such that the momentum map $\Phi \colon M \to \mathfrak{g}^*$ is $G$-equivariant w.r.t. the coadjoint-action on $\mathfrak{g}^*$. Assuming that $W := \Phi(M)$…
Olorin
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Hamiltonian vector field - confusion

I am trying to understand the definition of a Hamiltonian vector field. I take as my symplectic manifold the sphere $$p_1^2+p_2^2+p_3^2=1$$ with symplectic form $$\omega=p_1 dp_2 \wedge dp_3+ p_2 dp_3 \wedge dp_1 + p_3 dp_1 \wedge dp_2.$$ I take as…
Bob Johns
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Symplectic positive definite matrix.

I want to prove that any symmetric positive definite symplectic matrix, $A$, and any real number $\alpha >0$, also $A^{\alpha} \in \operatorname{Sp}(2n)$. I was given a hint to decompose $\mathbb{R}^{2n}$ into direct sum of $V_{\lambda}$ the…
MathematicalPhysicist
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When is symplectic pullback bundle trivial

Suppose $x : S \to M$ is a smooth map, where $M$ is a symplectic manifold and $S$ is a Riemann surface. Consider the pullback bundle $x^*TM \to S$. When is this bundle trivial (as symplectic vector bundle)? If $S$ has non-empty boundary then the…
user145907
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complex structure in odd-dimensional real vector space

If V is an odd-dimensional real vector space, then is there a linear map $J: V \to V$ satisfying $J^2=-1$? i.e. is there a complex structure in odd-dimensional real vector space?
Yui
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Understanding $r:\mathfrak{g}\rightarrow Vect(X)$ is the transpose of $d\mu:TX\rightarrow \mathfrak{g}^*$

I'm looking at a paper by S.K. Donaldson…
user138057
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tangent and conormal bundles of a Lagrangian

Suppose we have a Lagrangian submanifold $L$ of the symplectic manifold $T^*\mathbb{R}^{n}$ (endowed with symplectic form $\omega$), and a point $p\in L$. I know that there's a map $T_pL\rightarrow N^{*}_{p}L$, $X\mapsto\omega(X,\cdot)$. Why is…
Rod
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Complex structures on $T^4$

Suppose $(M,\omega)$ is a symplectic manifold, $J(M)$ is the space of all compatible complex structures. How can we show $J(T^4)$ is homotopic to the space of continuous maps $Map(T^4\rightarrow S^2)$?
user93417
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immersed symplectic surface VS immersed holomorphic curve

Let $(M, \omega)$ be a symplectic 4-manifold. An immersed symplectic surface is an immersion $v: S \to M$ such that $v^*\omega>0$ is a volume form. Fix a tame/compatible almost complex structure $J$. A holomorphic curve is a map $u:(\Sigma, j) \to…
seumath12345
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