a Lagrange multiplier
/ lah-GRAHNJ /
A clever bookkeeping device for optimizing while obeying a constraint. Instead of laboriously substituting the constraint in to eliminate a variable, you introduce one extra unknown per constraint and treat all the original variables as free. In mechanics that extra unknown turns out to be exactly the force of constraint.
To extremize something subject to f(q, t) = 0, you add a term lambda times f to what you vary and treat lambda as an independent variable. The Euler-Lagrange equations then become d/dt (partial L / partial q_dot_i) - partial L / partial q_i = lambda (partial f / partial q_i). The right-hand side is the generalized constraint force, and solving the system yields both the motion and the value of lambda. You need one multiplier per constraint equation.
You meet Lagrange multipliers throughout physics and optimization. In statistical mechanics, temperature and chemical potential appear as multipliers enforcing fixed average energy and particle number when you maximize entropy. In mechanics their advantage over simply eliminating a coordinate is that they hand you the constraint force, the tension or the normal force, as a free byproduct, which is often exactly what you want to know.
For a bead constrained to a wire, the multiplier lambda comes out as the reaction force the wire exerts on the bead.
One multiplier per constraint; its value is the constraint force.
The multiplier is not arbitrary: its value is fixed by the equations of motion, and it equals the force of constraint.