Lyapunov stability for some central forces
type: Item
instance of: scholarly article
title: Lyapunov stability for some central forces
zbMATH Open document ID: 0668.34051
DOI: 10.1016/0022-0396(88)90005-8
author: Gaetano Zampieri
published in: Journal of Differential Equations
publication date: 1988-01-01T00:00:00Z
review text: Consider the system (\(*\)): \(\ddot x+xf(x)=0\), \(\ddot y+yf(x)=0\), where \(f\in C^ 0(A,R)\), \(0\in A=\overset\circ A\subset \mathbb{R}\). If \(f(0)>0\) then the force is attractive and one could think that the stability always occurs as in the conservative case. This naive conjecture is false as proved by \textit{A. Barone} and the author in the paper Attractive central forces may yield Lyapunov instability. Proceed. VII ELAM, Equinoccio, Caracas, 1986, 105-112. They find more than only an example for this conjecture, and obtain large classes of f for which instability occurs and they completely solve the stability problem for f even: the equilibrium is then stable only in the trivial case, i.e. whenever f is locally constant at zero and \(f(0)>0\). Is stability possible just in the trivial case? The answer to this question is negative. The aim of this paper is to determine and construct all the continuous maps f such that the origin is a stable equilibrium for the system (\(*\)). This paper gives also a different proof of results on stability using an integral equation similar to Abel's equation, and introduce some results on the unstable cases.
Mathematics Subject Classification ID: 34D20
zbMATH DE Number: 4093954
zbMATH Keywords: central forces
zbMATH Keywords: Abel's equation
reviewed by: He Chungyou
MaRDI profile type: Publication
full work available at URL: 0022-0396(88)90005-8
OpenAlex ID: W2093185356
cites work: Q5540675
cites work: Non-conservative positional systems-stability
description: scientific article; zbMATH DE number 4093954