This paper can be regarded as a continuation of our previous work [1, 2] on the hierarchy of the dynamical behavior of nonholonomic systems. We consider different mechanical systems with nonholonomic constraints; in particular, we examine the existence of tensor invariants (laws of conservation) and their connection with the behavior of a system. Considerable attention is given to the possibility of conformally Hamiltonian representation of the equations of motion, which is mainly used for the integration of the considered systems.

Conservation laws, hierarchy of dynamics and explicit integration of nonholonomic systems

This article constitutes an appendix to the book by P. A. Griffiths, Exterior differential systems and the calculus of variations. Birkhauser, 1983. It especially focusses on the distinction between holonomic and nonholonomic mechanical and variational problems, and indicates how rich and interesting the phenomena are in the nonholonomic case.

Nonholonomic problems and the theory of distributions

In this historical review we describe in detail the main stages of the development of nonholonomic mechanics starting from the work of Earnshaw and Ferrers to the monograph of Yu. I.Neimark and N.A. Fufaev. In the appendix to this review we discuss the d’Alembert–Lagrange principle in nonholonomic mechanics and permutation relations.

Historical and critical review of the development of nonholonomic mechanics: the classical period

Some generalizations of the integration methods of the holonomic systems to the non-holonomic systems are given. The generalizations include that of the Whittaker method, the Poisson theory, the field method, the Noether symmetry and the Lie symmetry.

On the integration methods of non-holonomic dynamics

The conditions under which the three forms of Hamiloris principle were derived for nonholonomic systems with linear constraints by Holder, Voronets and Suslov are analysed in the general case of nonlinear constraints. It is proved, that these three forms are equivalent and transformable to each other. The analogous questions are analysed for the case of nonlinear quasi-coordinates and quasi-velocities. In addition the forms of Holder, Voronets and Suslov are exhibited in the case of Legendre transformation reducing the motion's equations to canonical f arm in quasi-coordinates. Also the conditions under which Hamilton's principle for nonholonomic systems has the characterises of the principle of stationary action are derived. .

Forms of Hamilton's principle for nonholonomic systems

The necessary and sufficient conditions of the applicability of the generalized Hamilton-Jacobi method for integrating the equations of motion of mechanical systems with nonholonomic constraints of Chetaev's type are obtained An example is considered



Es werden notwendige und hinreichende Bedingungen fur die Anwendbarkeit des verallgemeinerten Hamilton-Jacobischen Integrationsverfahrens auf die Bewegungsgleichungen mechanischer Systeme mit nichtholonomen Zwangsbedingungen vom Chetaevschen Typ erhalten Ein Beispiel wird betrachtet

V. V. Rumyantsev

Papers

On the Problem of a Generalization of the Hamilton‐Jacobi Method for Nonholonomic Systems