Nonholonomic mechanics: A practical application of the geometrical theory on fibred manifolds to a planimeter motion

Abstract

A geometrical theory of general nonholonomic mechanical systems on fibred manifolds and their jet prolongations, based on so-called Chetaev-type constraint forces, was developed in 1990s by Krupková. The relevance of this theory for general types of nonholonomic constraints, not only linear or affine ones, was then verified on appropriate models. Frequently considered constraints on real physical systems are based on rolling without sliding, i.e. they are holonomic, or semiholonomic, i.e. integrable. On the other hand, there exist some practical examples of systems subjected to true (non-integrable) nonholonomic constraint conditions. In this paper we study the planimeter—a mechanism for measuring areas which belongs to mechanical systems subjected to constraint conditions containing among others a true nonholonomic one.We study the planimeter motion using the above mentioned Krupková's approach. The results of numerical solutions of constrained equations of motion, derived within the theory, are in a good agreement with theoretical ones and thus they confirm the possibility of direct application of the theory to practical situations.