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symplectic
symplectic, a. and n. (sɪmˈplɛktɪk) [ad. Gr. συµπλεκτικός twining or plaiting together, copulative, f. σύν sym- + πλέκειν to twine, plait, weave: see -ic.] A. adj. 1. Anat. and Zool. Epithet of a bone of the suspensorium in the skull of fishes, between the hyomandibular and the quadrate bones.1839–4... Oxford English Dictionary
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Symplectic
In mathematics it may refer to: Symplectic Clifford algebra, see Weyl algebra Symplectic geometry Symplectic group Symplectic integrator Symplectic manifold Symplectic matrix Symplectic representation Symplectic vector space It can also refer to: Symplectic bone, a bone found in fish skulls wikipedia.org
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Symplectic integrator
symplectic methods do not apply. symplectic form. wikipedia.org
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Symplectic forms are isomorphic Let $V$ be a symplectic vector space, i.e. a vector space with a non-degenerate alternating bilinear form, a so-called symplectic form. There is a theorem: > All symplectic forms on $V...
It can be proven that any finite-dimensional symplectic vector space of dimension $2n$ has a basis in which the symplectic form $\omega$ has the form: $$\begin{pmatrix} 0 & I_n \\\ -I_n & 0 \end{pmatrix}$$ This proves that two symplectic forms are isomorphic.
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Symplectic group
Relationship with symplectic geometry Symplectic geometry is the study of symplectic manifolds. A symplectic vector space is itself a symplectic manifold. wikipedia.org
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Consider symplectic vector fields $X,Y$ and a symplectic connection $\nabla$. Is $\nabla_{X}Y$ symplectic? Consider a symplectic manifold $(M,\omega)$, together with a symplectic connection $\nabla$, _i.e_. a torsion-...
partial}{\partial y}$ (whose Hamiltonian function is $-x^2/2$), $Y = y \frac{\partial}{\partial x}$ (whose Hamiltonian function is $y^2 /2$) and the (symplectic Then $\nabla_X Y = x \frac{\partial}{\partial x}$, which is not symplectic as it is not Hamiltonian (or alternatively because it doesn't have vanishing
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Symplectic representation
(V, ω) which preserves the symplectic form ω. Representation theory Symplectic geometry wikipedia.org
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Open sets of symplectic manifolds Suppose I have a symplectic manifold $(\mathcal{M}, \omega)$. Does it hold that any open subset of $(\mathcal{M}, \omega)$ is a symplectic submanifold? The statement trivially holds ...
It trivially holds for symplectic manifolds as well. Let $(M^{2n}, \omega)$ be a symplectic $2n$-manifold and $X \subset M$ be an open subset. Therefore since $(T_x M, \omega_x)$ is a symplectic vector space by the fact that $(M,\omega)$ is a symplectic manifold, $(T_x X, \omega_x)$ is a symplectic
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Symplectic category
In mathematics, Weinstein's symplectic category is (roughly) a category whose objects are symplectic manifolds and whose morphisms are canonical relations See also Fourier integral operator Category theory Symplectic geometry wikipedia.org
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Example of symplectic manifold I wonder why tangent bundle is not symplectic. As you know, cotangent bundle is symplectic. (1) question 1 : Is cotangent bundle isometric to tangent bundle ? (2) question 2 :...
This implies, by transport of structure, that $TM$ can always be given the structure of a symplectic manifold. The issue is that while $T^*M$ is _canonically_ a symplectic manifold, $TM$ is not - the symplectic structure inherited from $T^\ast M$ depends on the
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Symplectic sum
In mathematics, specifically in symplectic geometry, the symplectic sum is a geometric modification on symplectic manifolds, which glues two given manifolds Around the same time, Eugene Lerman proposed the symplectic cut as a generalization of symplectic blow up and used it to study the symplectic quotient wikipedia.org
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Open immersion pulls back symplectic form to symplectic form? If $M$ is symplectic, and $f: W \to M$ is an open immersion, i.e. an immersion where $W$ and $M$ have the same dimension, does $f$ necessarily pull back a ...
A symplectic form is a 2-form $\omega$ on a $2n$-manifold such that $\omega^n$ is pointwise nonzero and $d\omega = 0$.
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Symplectic geometry
"Symplectic topology," which studies global properties of symplectic manifolds, is often used interchangeably with "symplectic geometry." A 2n-dimensional symplectic geometry is formed of pairs of directions in a 2n-dimensional manifold along with a symplectic form This symplectic form wikipedia.org
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为什么辛结构（Symplectic structures）在物理中非常有用？
最后提供一个网站，整理了symplectic structure in physics:http://www.phy.olemiss.edu/~luca/Topics/math/sympl_phys.html 如何理解symplectic potential减去哈密顿量这个形式呢？ （中间resolution of identity的时候使用角动量本征态即可保证角动量始终守恒） 这个看法有助于我们理解为什么拉格朗日力学无法通过直接代入守恒量来约化自由度，回忆拉氏量是由symplectic potential减去哈密顿量生成的，在约化（即限制到角动量守恒的路径）的过程中，symplectic zhihu
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Square root of Symplectic and Positive Definite Matrices in $M_{2n\times 2n}(\mathbb{R})$ Let $R$ be a symplectic matrix. Then the adjoint $R^*$ of $R$ is also a symplectic matrix. Then $RR^*$ is symplectic and positi...
$R^*$ is symplectic, so $J=R^*JR=U^*(AJA)U$, ie $AJA=UJU^*$, thus, with $J’=UJU^*$ orthogonal, we have $V:= AJ=J’A^{-1}$. Indeed, if the lemma holds, then $J=J’$ and $U$ is symplectic, therefore so is $A=RU^{-1}$.
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