A second order gradient representation of mechanics system

Abstract

A gradient representation and a second order gradient representation of the mechanics system are studied. The differential equations of motion of the holonomic and nonholonomic mechanics systems are expressed in the canonical coordinates. A condition under which the system can be considered as a gradient system is given. A condition under which the system can be considered as a second order gradient system is obtained. Two examples are given to illustrate the application of the result.

Stability analysis of a simple rheonomic nonholonomic constrained system

“…It is an important and difficult problem to study the stability for a nonholonomic constrained system. [1][2][3][4][5][6][7][8][9][10][11][12][13][14][15][16][17] It is also difficult to construct the Lyapunov function directly from the differential equations of the dynamical system. The authors in Ref.…”

Two kinds of generalized gradient representations for holonomic mechanical systems

“…Some results have been obtained in the research of the relation between constrained mechanical systems and gradient systems. [6][7][8][9][10][11][12][13][14][15][16][17][18] However, in those studies, the matrix or the function of the gradient system has no time t. When its matrix or function contains time t, the system can be called a generalized gradient system. There are two types of generalized gradient systems that are particularly useful for the study of stability.…”