On the Equations of Motion for Constrained Systems and Their Stability

Abstract This paper presents recent advances and reviews several notions on constrained systems that have evolved over the years. It questions several of the ‘received (conventional) views’ on both holonomic and non-holonomic systems and shows their erroneous nature in light of recent developments. By using simple illustrative examples, it draws attention to several key ideas that differ from conventional views on the behavior of constrained systems, and especially their stability. Using the explicit equation of motion for constrained systems, we show that an degree of freedom unconstrained dynamical system upon which holonomic and/or nonholonomic constraints are imposed, remains an n degree of freedom system with the initial conditions allowed by the constraints. A central result on the stability of systems using the explicit equation is developed. It encompasses holonomic and/or nonholonomic constraints, including a wide class of nonlinear nonholonomic constraints. Several commonly held notions related to constrained systems are examined and found to require refinement, alteration, or rejection.