Lyapunov stability for some central forces
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.
scientific article; zbMATH DE number 4093954