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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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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Proving that a particular submanifold of the cotangent space is Lagrangian
I have the following problem in my differential topology class: Let $M$ be an $n$-dimensional manifold and let $\omega$ denote the standard symplectic form on the cotangent space $T^*M$. Let $f \in ...
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Why is $Sp(2m)$ as regular set of $f(A)=A^tJA-J$, and, hence a Lie group.
I'm trying to prove that $Sp(2m)$ is a Lie group using this: defining a function $f(A)=A^tJA-J$ and trying to see that this is a submersion. But I've not realized yet what is the domains and the range ...
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Symplectic submanifolds and first integrals
I was working with symplectic submanifolds when I posed the following question:
Suppose I have a Hamiltonian system with the phase space $\mathcal{M}$, a symplectic manifold with the standard ...
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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= $ ...
