A geometric approach to constraint stabilization for holonomic lagrangian systems

In this paper, we develop a geometric approach to constraint stabilization for holonomic mechanical systems in the context of Lagrangian formulation. We first show that holonomic mechanical systems, for the case in which a given Lagrangian is hyperregular, can be formulated by using the Lagrangian two-form, namely, a symplectic structure on the tangent bundle of a configuration manifold that is induced from the cotangent bundle via the Legendre transformation. Then, we present an idea of geometric constraint stabilization and we show that a holonomic Lagrangian system with geometric constraint stabilization can be formulated by the Lagrange-d'Alembert principle, together with its local coordinate expression for the sake of numerical computations. Finally, we illustrate the numerical verification that the proposed method enables to stabilize constraint violations effectively in comparison with the Baumgarte and Gear-Gupta-Leimkuhler methods together with an example of a linkage mechanism.