Ciro Santilli $$Sponsor Ciro$$ 中国独裁统治 China Dictatorship 新疆改造中心、六四事件、法轮功、郝海东、709大抓捕、2015巴拿马文件 邓家贵、低端人口、西藏骚乱
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# Symplectic group ()

| nosplit | ↑ parent "Classical group" | words: 222 | descendant words: 222 | descendants: 2
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Intuition, please? Example? https://mathoverflow.net/questions/278641/intuition-for-symplectic-groups The key motivation seems to be related to Hamiltonian mechanics. The two arguments of the bilinear form correspond to each set of variables in Hamiltonian mechanics: the generalized positions and generalized momentums, which appear in the same number each.
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Seems to be set of matrices that preserve a skew-symmetric bilinear form, which is comparable to the orthogonal group, which preserves a symmetric bilinear form. More precisely, the orthogonal group has: $$OTIO=I (185)$$ and its generalization the indefinite orthogonal group has: $$OTSO=I (186)$$ where S is symmetric. So for the symplectic group we have matrices Y such as: $$YTAY=I (187)$$ where A is antisymmetric. This is explained at: https://www.ucl.ac.uk/~ucahad0/7302_handout_13.pdf They also explain there that unlike as in the analogous orthogonal group, that definition ends up excluding determinant -1 automatically.
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Therefore, just like the special orthogonal group, the symplectic group is also a subgroup of the special linear group.
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