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.

About this 'force', could someone explain/show to me

  • 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.

Regarding your questions:

a. Why cannot it do work?

This of course is equation-dependent, however if your equation is relevant to a mechanical constraint, this constraint will express some impenetrability/incompressibility, so that velocity/deformation will have to be zero in the direction of the force that arises from the Lagrange multiplier. Thus work will be zero, which is consistent with the fact that you express a constraint and not an energy source.

b. What happens when the constraint is redundant, i.e. already satisfied?

That's simpler: the Lagrange multiplier will be trivial (zero if it is a force, or a constant in the case of pressure), such that the resulting force will be zero.

  • $\begingroup$ One might add that the deformation along the gradient of the constraining function is zero because that direction is orthogonal to the constrained submanifold, and there is therefore no motion in that direction as long as the constraint is satisfied. $\endgroup$ – Emilio Pisanty Apr 22 '14 at 7:05

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