A consistent formulation in terms of generalized coordinates is presented for elastic, conservative systems with kinematic constraints. Constraint functions are defined and formulated together with the total potential energy of the system; this leads to equilibrium conditions in terms of variational inequalities. In the present approach, the equilibrium conditions are rewritten in the form of the Kuhn-Tucker conditions. Next, a perturbation analysis is carried out to investigate uniqueness of the solution and to distinguish between regular and critical points. The authors develop two ways to classify a state. In the first approach, a matrix is defined in terms of the Hessian of the Lagrangian and the gradient of the active constraints. In the second approach, the projection of the Hessian of the Lagrangian on a tangent space is employed.
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