Programmed motion of mechanical systems

Abstract

The main results of this contribution are new methods of solving of adjoint systems of 6n di erential nonlinear equations and its application in classical and celestial mechanics. At first, we are observing a general dynamic system of n differential equations of the first order, which contain n independent functions x(t) and n unknown composite functions X(x(t)). A programme of motion of such one is described by n independent finite algebraic equations f(x) = 0. For realization of the control motion it is necessary to de ne functions X(x) and within them also control functions. It is shown that such dynamic systems do not correspond to mechanical systems. Defining of control motion of mechanical systems is much more complex. It is explained which of the di erential equations of motion are used, and what are the consequences. It is also manifested that 3N Newton's differential equations of motions and n = 3N−k, k < 3N, Lagrange's differential equations of second kind, or 2n Hamilton's differential equations on manifolds, are not giving the same results at defining of forces, being of the primary importance for control motion.