To find the extrema of a function

, subject to the constraint that

, form an auxiliary function

. The coordinates

of the extrema satisfy three equations:

,

, and

=0. The function

is referred to as the Lagrangian, and

as the Lagrange multiplier. One observes that the first of the three equations is simply

, and that we really have three variables

, so the system we solve is not overdetermined. This method assumes that the constraint equation defines a smooth (differentiable) closed curve, so that no points require special testing.