Symplectic leaves

symplectic geometry

‹william›The symplectic leaves are equivalence relations \(x \tilde y\) if and only if \(x\) can be connected to \(y\) be a piece-wise Hamiltonian path 27/02/201123:12:31‹william›\(x \sim y\)27/02/201123:14:18‹william›Let \(D\) be a degenerate distribution27/02/201123:14:49‹william›this means that for every point \(x \in M\), \(D_x\) is a subset of \(T_x M\)27/02/201123:14:58‹william›subset = subspace27/02/201123:15:18‹william›distribution normally means that \(D_x\) is constant rank27/02/201123:15:25‹william›and \(D_x\) is spanned by vector fields27/02/201123:15:58‹william›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 \(D_x\)27/02/201123:16:19‹william›and \(X_1(y),\ldots,X_r(y)\) lie in \(D_y\) for all \(y\) where they are defined27/02/201123:18:02‹william›a foliation of \(D\) is an immersed manifold \(A\) of \(M\) with \(TA = D\)27/02/201123:19:10‹william›Let \(M^{dis}\) be the manifold with underlying set \(M\) and the discrete topology27/02/201123:20:09‹william›\(M^{dis}\) is an immersed manifold for \(D = M \times 0\)27/02/201123:20:36‹william›\(M = \R^2\)27/02/201123:20:46‹william›\(D = \R^2 \times \R\)27/02/201123:24:17‹william›the foliation is the map \(\bigcup_{\R} \R \arr \R^2\)27/02/201123:24:25‹william›the foliation is the map \(\bigcup_{\R} \R \rightarrow \R^2\)