Study on the Core Groups of First Integrals and Folding Index for Mechanical Systems

Abstract

In applying the Udwadia–Kalaba equation for constrained mechanical systems, a direct proof of the equivalence of first integrals and nonholonomic constraints is given, and it is demonstrated that the generalized force of the system is equivalent to the constraint force derived by all first integrals of the nonholonomic constraints. Furthermore, depending on whether complete information is included in the subsets of the first integrals or not, the concept of “multiple kernel” of the system is introduced, and then the core groups of the first integrals and the folding index, which reveals the “simplicity” of the system, are defined. Finally, the onefold system is discussed in detail, and the judgment method is given. To verify the feasibility of this method and illustrate the application of the multiple kernel theory, three examples are considered. The new concepts and results presented in this paper help reveal the inner structure of the general mechanical system, which forms the foundation of control theory of constraint motions, and the multiple kernel analysis of the complex systems can be a new research area of analytic mechanics in the future.