Symplectic leaves

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 \(\bigcup_{\R} \R \rightarrow \R^2\)

related items

• Foliation dynamics