Stability preserving structural transformations of systems of linear second-order ordinary differential equations

TL;DR

This paper develops a theory of stability-preserving transformations for second-order linear ODE systems, generalizing classical theorems and applying them to analyze the stability of a rigid body's rotary motion.

Contribution

It introduces a new theoretical framework for stability-preserving transformations of second-order ODE systems, extending classical stability theorems.

Findings

Generalized Kelvin--Tait--Chetayev theorems

Applied theory to rigid body rotary motion

Established stability criteria for specific systems

Abstract

In the paper we have developed a theory of stability preserving structural transformations of systems of second-order ordinary differential equations (ODEs), i.e., the transformations which preserve the property of Lyapunov stability. The main Theorem proved in the paper can be viewed as an analogous of the Erugin's theorem for the systems of second-order ODEs. The Theorem allowed us to generalize the 3-rd and 4-th Kelvin -- Tait -- Chetayev theorems. The obtained theoretical results were successfully applied to the stability investigation of the rotary motion of a rigid body suspended on a string.