tailieunhanh - The problem of first integral of nonholonomicsystems

In this paper the problem of first integrals of a. nonholonomic system is discussed. The aim of this work is concentrated on finding the condition for existence of first integrals. The obtained results are applied for the construction of linear and quadratic integrals of nonholonomic systems. | Vietnam Journal of Mechanics, NCST of Vietnam Vol. 23, 2001, No 1 (51 - 64) THE PROBLEM OF FIRST INTEGRAL OF NONHOLONOMICSYSTEMS Do SANH Hanoi University of Technology ABSTRACT. In this the problem of first integrals of a. nonholonomic system is discussed. The of this work is ed on finding the condition for existence of first integrals. The obtained results are applied for the construction of linear and quadratic integrals of nonholonomic systems. It obtains two important affirmations; they are: Any first integral could be treated as a particular nonholonomic constraint and contrarily, any nonholonomic constraint could be regarded as a first integral of the nonholonomic system. 1. Introduction The construction of first int egrals of nonholonomic systems is one of the hardest problems of dynamics of nonholonomic systems, due to the presence of nonholonomic constraints. There is also no convenient form of equations of motion that describe these systems. It is known presently the equations of motion applied for nonholonomic systems are either the Lagrange's or the Appell's equations. In the first case the unknowns are increased due to introducing multipliers, but in the latter, the calculation of the acceleration energy is difficult. Investigations about this field have been done very few. In order to overcome the above mentioned restrictions, the form of equations of motion for a nonholonomic systems presented in [8) is applied. The form of these equations is analogous to one equation which describe,s the motion of a nonconser. vative holonomic system. 2. The condition of of a first integral for nonholonomic systems Let us consider a dynamic system, which is described by the equations () where xi are the st ate variables, ·but Xi = dti . dx · As known (see, for example [9], [17]) in order the relation () 51 is a first, integral of () the necessary and sufficient condition is Of :::, I:.g(x1,x2, . ,xn) .