On Hamilton's principle for nonholonomic systems

Abstract

The conditions under which the three forms of Hamilton's variational pricciple were derived for nonholonomic systems by Holder [1], Voronets [2], and Suslov [3] are analyzed in the general case of nonlinear and, also, in particular cases of linear relationships. It is shown that these three forms are equivalent and transformable into one another. Gererally Hamilton's principle in relation to nonholonomic systems is not the principle of stationary action. although under specific conditions of real motion of such systems it can be found among solutions of Euler's equations of the Lagrange variational problem. The conditions under which Hamilton's principle applied to related motions of a nonholonomic system has the characteristics of the principle of stationary motion are derived. This is closely related to the question of applacability to nonholonomic systems of the generalized Hamilton-Jacobi method of integrating the equations of motion [4], The necessary and sufficient conditions of that method applicability to nonholonomic systems have been found to be equivalent to the conditions noted above [5]. It is shown that the method is applicable then and only then when Hamilton's principle can be treated as the principle of stationary action. Examples are presented.