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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Showing existence of symplectic transformations preserving a quadratic form

Question: I need help to prove the following statement. Let $W_i:=w_iw_i^T\in\mathbb{R}^{n\times n}$, for $n$ even, be symmetric rank-1 matrices, $J=-J^T$ the canonical symplectic matrix and define ...
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Bounding norms of symplectic matrix factorisations and non-seperable Hamiltonian flows

Problem setup: Let $e^{hJM}$ be the time-$h$ flow corresponding to the ODE $\dot{x} = JMx$, with $M = \left(\begin{array}{cc} A & C\\ C^T & B\\ \end{array}\right)$ symmetric positive ...
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$Ham(M, \omega)$ acts transitively on $(M,\omega)$

Let $M$ be a compact and connected smooth manifold with a symplectic form $\omega$. $Ham(M, \omega)$ denotes the space of hamiltonian symplectomorphisms of $(M,\omega)$. I have the following ...
• 589
Let $M$ a compact and connected smooth manifold. Suppose $X_t$ is a time-dependent symplectic vector field and let $\phi_t$ be the isotopy generated by $X_t$. Prove that $\phi_t ∈ Symp(M, \omega)$ for ...