Mathematical Problems in Engineering
Volume 2012 (2012), Article ID 543098, 32 pages
Research Article

Theoretical Analysis for a Class of Rheonomous Affine Constraints on Configuration Manifolds—Part I: Fundamental Properties and Integrability/Nonintegrability Conditions

Department of Applied Electronics, Faculty of Industrial Science and Technology, Tokyo University of Science, Chiba 278-8510, Japan

Received 6 May 2012; Accepted 4 July 2012

Academic Editor: Wei-Chiang Hong

Copyright © 2012 Tatsuya Kai. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.


We analyze a class of rheonomous affine constraints defined on configuration manifolds from the viewpoint of integrability/nonintegrability. First, we give the definition of A-rheonomous affine constraints and introduce, geometric representation their. Some fundamental properties of the A-rheonomous affine constrains are also derived. We next define the rheonomous bracket and derive some necessary and sufficient conditions on the respective three cases: complete integrability, partial integrability, and complete nonintegrability for the A-rheonomous affine constrains. Then, we apply the integrability/nonintegrability conditions to some physical examples in order to confirm the effectiveness of our new results.