The Sturm theorems and symplectic geometry (Q1084720)

The goal in a more topological context of this paper is to rephrase the classical theorems of Sturm on oscillation and non-oscillation of solutions to a second order differential equation. For the higher dimensional system \[ \ddot x=-A(f)x,\quad x\in R^ n,\quad A^ Y=A \] the important concept is the evolution of a Lagrangian plane in the symplectic phase space. The natural generalization of the classical case giving zeros of x(t) is the search for times when a given Lagrangian plane becomes vertical, i.e., not transverse to the fibers of the cotangent bundle. The author presents the analogue of the classical Sturm theorems in this context.