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On the equations of motion of mechanical systems subject to nonlinear nonholonomic constraints Ghori, Qamaruddin Khan

Abstract

Suppose q₁,q₂,…,qn are the generalised coordinates of a mechanical system moving with constraints expressed by r non-integrable equations (r〈n) (1) [equation omitted] where the dots denote differentiation with respect to the time t, and fα are nonlinear in the q’s. The equations (1) are said to represent nonlinear nonholonomic constraints and the system moving with such constraints is called nonlinear nonholonomic. From a purely analytical point of view, the author has obtained the equations of motion for a nonlinear nonholonomic mechanical system in many a different form. The importance of these forms lies in their simplicity and novelty. Some of these forms are deduced from the principle of d'Alembert-Lagrange using the definition of virtual (possible) displacements due to Četaev [ll] . The others are obtained as a result of certain transformations. Moreover, these different forms of equations of motion are written either in terms of the generalised coordinates or in terms of nonlinear nonholonomic coordinates introduced by V.S. Novoselov . These forms involve the energy of acceleration of the system or the kinetic energy or some new functions depending upon the kinetic energy of the system. Two of these new functions, denoted by R (Sec. 2.3) and K (Sec. 2.4), can be identified, to a certain approximation, with the energy of acceleration of the system and the Gaussian constraint, respectively. An alternative proof (Sec.2.5) is given to the fact that, if virtual displacements are defined in the sense of N.G. Četaev [ll], the two fundamental principles of analytical dynamics - the principle of d'Alembert-Lagrange and the principle of least constraint of Gauss -are consistent. If the1 constraints are rheonomic but linear, a generalisation of the classical theorem of Poisson is obtained in terms of quasi-coordinates and the generalised Poisson's brackets introduced by V.V. Dobronravov . The advantage of the various novel forms for the equations of motion is illustrated by solving a few problems.