Formulation and Solution of a Generalized Chebyshev Problem: First Part

Abstract

This paper is devoted to studying the motion of non-holonomic systems with higher-order constraints. The problem of the motion of such systems is formulated as the generalized Chebyshev problem. This refers to the problem in which the solution to a system of equations of motion should simultaneously satisfy an auxiliary system of higher-order (n\( \geqslant \) 3) differential equations. Two theories are constructed to study the motion of these systems. In the first, a joint system of differential equations for the unknown generalized coordinates and Lagrange multipliers is constructed. In the second theory, the equations of motion are derived by applying the generalized Gauss principle. The higher-order constraints are considered the program constraints in this investigation. Thus, the problem of finding the control satisfying the program given in the form of auxiliary system of differential equations linear in the (n\( \geqslant \) 3)-order derivatives of the sought generalized coordinates is formulated. A novel class of control problems is therefore introduced into consideration. Several examples are provided of solving the real mechanical problems formulated as the generalized Chebyshev problems. The paper is a review of the research performed for many years at the Department of Theoretical and Applied Mechanics of St. Petersburg University.