on a contour line or (the constant is unknown, as the contour line is what we are looking for) tangent to the previous one.Д«y imposing (for any ), we impose that and share the same orientation but with different relative slopes (depending on ). The global problem can be understood as finding a point that is:Дў. This in turn shows how the independence of and is restored by the introduction of (which is adjusted so that and are strictly parallel in both slope and orientation, so that their height difference remains constant near the solution, with no more constraint between and ). This Demonstration illustrates the 2D case, where in particular, the Lagrange multiplier is shown to modify not only the relative slopes of the function to be minimized and the rescaled constraint (which was already shown in the 1D case), but also their relative orientations (which do not exist in the 1D case). See Lagrange Multipliers in One Dimension. The main ideas behind the Lagrange multipliers have already been discussed in the 1D case. The equilibrium condition ( ) can be seen as a constraint on. The function (plotted in green) is the difference in height between and the rescaled constraint function, which is constant only when and are parallel.

Hence, as, for obvious symmetry reasons the complete solution here is quite simple. In this particular case, the new potential is a Legendre transformation, where (given the constraint ). Here, so the constraint is, which is simpler to visualize (thick blue line).

The constraint is its particular contour line. The function is the constraint function, plotted in blue. The function (plotted in red) is the one to be optimized subject to the constraint.

Wolfram mathematica 11.3