The Sturm theorems and symplectic geometry
type: Item
instance of: scholarly article
title: The Sturm theorems and symplectic geometry
zbMATH Open document ID: 0606.58017
DOI: 10.1007/BF01077289
publication date: 1985-01-01T00:00:00Z
review text: 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.
Mathematics Subject Classification ID: 37J99
Mathematics Subject Classification ID: 34C10
zbMATH DE Number: 3980066
zbMATH Keywords: theorems of Sturm
zbMATH Keywords: oscillation
zbMATH Keywords: non-oscillation
zbMATH Keywords: Lagrangian plane
author: Vładimir I. Arnold
MaRDI profile type: Publication
cites work: On the iteration of closed geodesics and the sturm intersection theory
cites work: A generalized Sturm theorem
cites work: Q3262807
cites work: Invariant elements in modular lattices
cites work: Q3961757
cites work: Q4760460
full work available at URL: https://doi.org/10.1007/bf01077289
OpenAlex ID: W2064839383
published in: Functional Analysis and Its Applications
description: scientific article; zbMATH DE number 3980066