"Symplectic leaves"의 두 판 사이의 차이
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imported>Pythagoras0 |
imported>Pythagoras0 |
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| (같은 사용자의 중간 판 하나는 보이지 않습니다) | |||
| 16번째 줄: | 16번째 줄: | ||
* the foliation is the map <math>\bigcup_{\R} \R \rightarrow \R^2</math> | * the foliation is the map <math>\bigcup_{\R} \R \rightarrow \R^2</math> | ||
| + | ==related items== | ||
| + | * [[Foliation dynamics]] | ||
[[분류:개인노트]] | [[분류:개인노트]] | ||
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[[분류:math and physics]] | [[분류:math and physics]] | ||
[[분류:classical mechanics]] | [[분류:classical mechanics]] | ||
| + | [[분류:migrate]] | ||
2020년 11월 13일 (금) 07:47 기준 최신판
introduction
- symplectic geometry
- The symplectic leaves are equivalence relations \(x \sim y\) if and only if \(x\) can be connected to \(y\) be a piece-wise Hamiltonian path
- Let \(D\) be a degenerate distribution
- this means that for every point \(x \in M\), \(D_x\) is a subset of \(T_x M\)
- subset = subspace
- distribution normally means that \(D_x\) is constant rank
- and \(D_x\) is spanned by vector fields
- which means that for every \(x\) there is vector fields \(X_1,\ldots,X_r\) locally defined around \(x\) such that \(X_1(x),\ldots,X_r(x)\) span \(D_x\)
- and \(X_1(y),\ldots,X_r(y)\) lie in \(D_y\) for all \(y\) where they are defined
- a foliation of \(D\) is an immersed manifold \(A\) of \(M\) with \(TA = D\)
- Let \(M^{dis}\) be the manifold with underlying set \(M\) and the discrete topology
- \(M^{dis}\) is an immersed manifold for \(D = M \times 0\)
- \(M = \R^2\)
- \(D = \R^2 \times \R\)
- the foliation is the map \(\bigcup_{\R} \R \rightarrow \R^2\)