"Symplectic leaves"의 두 판 사이의 차이
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imported>Pythagoras0 |
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| − | [[symplectic geometry | + | ==introduction== |
| + | *[[symplectic geometry]] | ||
| + | * The symplectic leaves are equivalence relations <math>x \sim y</math> if and only if <math>x</math> can be connected to <math>y</math> be a piece-wise Hamiltonian path | ||
| + | * Let <math>D</math> be a degenerate distribution | ||
| + | * this means that for every point <math>x \in M</math>, <math>D_x</math> is a subset of <math>T_x M</math> | ||
| + | * subset = subspace | ||
| + | * distribution normally means that <math>D_x</math> is constant rank | ||
| + | * and <math>D_x</math> is spanned by vector fields | ||
| + | * which means that for every <math>x</math> there is vector fields <math>X_1,\ldots,X_r</math> locally defined around <math>x</math> such that <math>X_1(x),\ldots,X_r(x)</math> span <math>D_x</math> | ||
| + | * and <math>X_1(y),\ldots,X_r(y)</math> lie in <math>D_y</math> for all <math>y</math> where they are defined | ||
| + | * a foliation of <math>D</math> is an immersed manifold <math>A</math> of <math>M</math> with <math>TA = D</math> | ||
| + | * Let <math>M^{dis}</math> be the manifold with underlying set <math>M</math> and the discrete topology | ||
| + | * <math>M^{dis}</math> is an immersed manifold for <math>D = M \times 0</math> | ||
| + | * <math>M = \R^2</math> | ||
| + | * <math>D = \R^2 \times \R</math> | ||
| + | * the foliation is the map <math>\bigcup_{\R} \R \rightarrow \R^2</math> | ||
| − | + | ==related items== | |
| − | + | * [[Foliation dynamics]] | |
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| − | + | [[분류:개인노트]] | |
| − | + | [[분류:physics]] | |
| − | + | [[분류:math and physics]] | |
| − | + | [[분류:classical mechanics]] | |
| − | + | [[분류:migrate]] | |
2020년 11월 13일 (금) 06:47 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 \(\bigcup_{\R} \R \rightarrow \R^2\)