Another Look at Lagrange Multipliers

  Lagrange introduced the Method of Multipliers in his Analytical Mechanics of 1811 but when one first encounters this method it's not very clear why it works. One can deduce the procedure starting with a set of condition equations, Φ, and use least squares.

One ends up with a linear combination of the condition equations with arbitrary coefficients, the dΦ, set equal to zero. Division by dΦ₁ removes some of the arbitrariness since the linear combination equals a constant, -Φ₁.

Edit (Mar 20): Add dΦ and last sentence.