A Duality Principle in Nonholonomic Mechanical Systems

Abstract

The paper presents a duality principle in nonholonomic mechanical systems extensively utilizing Birkhoff’s idea of nonenergicness. It is shown that the nonenergic condition provides a structure of constrained mechanical systems specifically focusing on the nonholonomic systems, and that a duality principle is employed by analogy with Planck-Okada-Arsove’s theorem in network theory. Introducing connection matrices, a dual pair of constraints for velocities and forces is developed in a causal form to formulate required kinematical and dynamical relations associated with ideal constraints, and geometric properties of configuration subspaces are also investigated. It is finally demonstrated how reduced dynamical equations of the nonholonomic systems are derived with adjoint kinematical equations by exploiting the duality principle together with an illustrative example.