Theoretical analysis for a class of rheonomous affine constraints on configuration manifoldsPart I: Fundamental properties and integrability/ nonintegrability conditions

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.