tailieunhanh - Canonical equations for a constrained mechanical system

In the present work the author proposes a form of canonical equations for a constrained mechanical system applying usefully for holonomic and nonholonomic systems. These equations are constructed by the help of the principle of compatibility [1]. Such a form of canonical equations will be used comfortable for studying dynamic of a multibody system. | Journal of Mechanics, NCNST of Vietnam T. XVI, 1994, No 1 {43- 48) CANONICAL EQUATIONS FOR A CONSTRAINED MECHANICAL SYSTEM DO SANH Hanoi Technology Univers£ty §1. INTRODUCTION In many theoretical studies it is convenient to transform Lagrange's equations to the canonical form where the canonical variables are introduced for substuting the Lagrange's ones. It is a set of 2n variables { qi, pi} (i = 1, n) and in these variables the motion of a system is described by 2n ordinary differential equations of the first order. First, as known the cano~ical equations was established for a conservative holomonic mechanical system, Late~ a similar form was expended for a nonconservative mechanical system and nextly, for a nonholonomic system (the form of canonical equations with undefined multipliers} [2, 3, 8]. However, the above mentioned estsblished form of canonical equations haven't many practice senses. In the present work the author proposes a form of canonical equations for a constrained mechanical system applying usefully for holonomi_c and nonholonomic systems. These equations are constructed by the help of the principle of compatibility [1]. Such a form of canonical equations will be used comfortable for studying dynamic of a multibody system. §2. CANONICAL EQUATIONS FOR A CONSTRAINED MECHANICAL SYSTEM Let us consider a holonomic mechanical system. The position of the system is defined by Lagrange's coordinates qi (i = ~). There exists a force function U of active forces. Hamilton reduced the differential equations of motion to a very significant form called the Hamilton canonical equations. For the aim of establishing canonical equations, instead of variables qi we introduced new variables Pi (i = 1, n), that is: 8T Pi = aqi () where T is the kinetic energy of. the system which is assumed to be positive define quadratic form. The variables Pi are known as impulses and are cOnjugates of the Lagrange's coordinates. Since the highest order of form with .