The instability of the equilibrium of a two-degree-of-freedom gyro
type: Item
instance of: scholarly article
title: The instability of the equilibrium of a two-degree-of-freedom gyro
zbMATH Open document ID: 0661.70014
published in: Moscow University Mechanics Bulletin
publication date: 1987-01-01T00:00:00Z
review text: Consider an analytical mechanical system with the Lagrangian function \[ (1)\quad L=(\dot q_ 1^ 2;\dot q^ 2_ 2)+\omega (q_ 1\dot q_ 2- q_ 2\dot q_ 1)+V(q_ 1,q_ 2). \] Lagrangian equations with the Lagrangian (1) can be, for example, equations of motion of a material point on a plane, acted upon by certain potential forces and a magnetic force, with the vector of magnetic induction perpendicular to the plane of motion. Let \(q_ 1=q_ 2=0\) be the equilibrium position of the system, and suppose that the reduced force function V can be expanded as a MacLaurin series \(V(q_ 1,q_ 2)=\sum^{\infty}_{k=0}V_{m+k}(q_ 1,q_ 2),\) \(m\geq 2\), where \(V_{m+k}\) are homogeneous polynomials of degree \(m+k\). If \(\omega =0\) and the form \(V_ m\) can take positive values, the equilibrium position \(q_ 1=q_ 2=0\) is unstable. If, however, \(\omega\neq 0\), the gyroscopic stabilization is possible, and it is of interest to investigate sufficient conditions of instabilities, some of these conditions are stated and proved in this paper.
Mathematics Subject Classification ID: 70E05
Mathematics Subject Classification ID: 70K20
zbMATH DE Number: 4081004
zbMATH Keywords: analytical mechanical system
zbMATH Keywords: Lagrangian function
zbMATH Keywords: MacLaurin series
zbMATH Keywords: gyroscopic stabilization
zbMATH Keywords: sufficient conditions of instabilities
author: Stanislav D. Furta
MaRDI profile type: Publication
description: scientific article; zbMATH DE number 4081004