# Interpreting Lagrange Multipliers as forces

I am (still) working on getting a good understanding of Lagrange multipliers. I understand their function in an optimization problem that is subject to some constraint.

For the specific case of equations of motion of a dynamic system (Newton-Euler equations), my teacher showed me that the mathematical multiplier can physically be interpreted as constraint force.

• why it never does/cannot do work? (Wikipedia)

• what happens when the constraint is redundant, i.e. already satisfied?

This can be understood in the general case of Lagrange multipliers in mechanical systems. First, let me stress that in general, the Lagrange multiplier does not have to be a force. In the case of simple action-reaction, it is, but it is e.g. the gradient of pressure which is a force in the case of incompressible fluids.