Vol.5, No 1, 2006 pp. 43 - 51
UDC 531.011
ON INVESTIGATION OF DYNAMICAL SYSTEMS
WITH CONSTRAINTS
Viktor G. Veretennikov, Pavel S. Krasilnikov, Valery A. Sinictsyn
Department of Theoretical Mechanics, Moscow Aviation Institute
(State Technical University), Moscow, 125871, Russia,
e-mail: kras@k804.mainet.msk.su
Abstract. The dynamical systems with constraints (differential-algebraic systems) are investigated by methods of analytical mechanics. So, the well-known mechanical principle of release from constraints is extended to DAE systems. The definition of ideal constraint is formulated for these systems. It is shown that a necessary and sufficient condition for constraint forces to have a representation by Lagrange's multipliers is that the constraint be ideal. It is obtained the condition of ideality for the constraint dependents on the method of physical realization of restriction. Therefore one and the same constraint may be as ideal so nonideal. The examples are considered. The principal equation for dynamical systems with ideal constraints is obtained. For Chetaev's systems, the principal equation is also derived. As an example, the problem of the construction of periodic solutions for average Lorenz's dynamical system is considered.
O ISTRAŽIVANJU DIMAMIČKIH SISTEMA
POD DEJSTVOM VEZA
Dinamički sistemi pod dejstvom veza (diferencijalni algebarski sistemi DAES) istražuju se metodama analitičke mehanike. Tako, dobro poznati princip mehanike - oslobadjanja od dejstva veza proširen je na DAE sistema. Definicija idealnih veza je formulisana za ove sisteme. Pokazuje se da su potrebni i dovoljni uslovi da da bi sile otpora veza bile predstavljene Lagranžeovim  množiocima veza da su veze idealne. Došlo se do uslova idealnosti veza koji zavisi od metode fizičke realizacije veza. Stoga jedna i ista veza može da bude idealna i neidelna. Razmatreni su primeri. Princip jednakosti za dinamičke sisteme sa idealnim vezama se dobija. Za Četajeve sisteme, princip jednakosti je takodje izveden. Kao primer je proučen problem konstrukcije periodičnih rešenja za usrednjen Lorencov dinamički sistem.