The effects of constraints imposed on mechanical systems may be analyzed when the constraints are recognized as geometrical surfaces. Each component constraint restricts the motion due to the applied active forces; taken together the constraint surfaces intersect, forming a new dimensionally reduced surface and thus a single constraint. The gradient vectors on each of the component constraint surfaces must sum to the gradient on the reduced surface. Lagrange multipliers are then just scalars which adjust the magnitudes and senses of the gradient vectors on each component geometrical surface so that the gradient on their intersection has the proper magnitude and is directed along the resultant reaction force. Since the resultant gradient lies in a subspace with a basis formed by the component gradients, these component gradients must be linearly independent. The existence of a nonsingular Jacobian transformation from the subspace to the constraint surfaces guarantees its existence with the required dimension, and thus, the validity of the constraints is guaranteed.

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