The initial motions for holonomic and nonholonomic mechanical systems

Abstract

Whittaker first put forward a new approach, called the initial motions, to solve the differential equations of motion aimed at holonomic systems. Since most of the differential equations of motion for mechanical systems are nonlinear ordinary ones, which are difficult to find the analytic solutions. Fortunately, the concept of initial motions can manage these situations and study its subsequent motions. This work is devoted to discuss and investigate the initial motions for mechanical systems, particularly for nonholonomic systems. First, the differential equations for holonomic systems are formulated, and the formulation and solution of initial motions of the systems are proposed. Second, the differential equations of motion for nonholonomic systems are established, based on the new method of initial motions to obtain the initial values of high-order derivatives of generalized velocities, the formulation and solution of initial motions are introduced in the general nonholonomic systems and Chaplygin systems. The methods and results obtained are illustrated by a number of classical examples, both for holonomic and nonholonomic systems.