Lagrange multipliers and the Indian paper
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A short personal note by Phil dated 9.20.16, in a folder of Goldstein-related mechanics files. He tries to see whether his own constrained-extremum argument connects to Section IV of another paper, since both use s constraints and s multipliers. He asks why setting all M+s partial derivatives of H = F + Σλf to zero is justified, relating it to linear dependence of the rows of his R matrix and his Theorem 1 on stationary points.
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Lagrange multipliers and the Indian paper PhL 9.20.16
I understand my paper by itself, and I understand Section IV of the Indian paper by itself. Here I want to figure out if there is some connection between them that I am not grokking. Both papers deal with s constraints, and both have s multiplier factors λi. Call these papers P and I.
P: I want an extremum of F(x1.....xM). If the variables were independent, I would search for points where all M equations ∂iF = 0. But the equations are not independent due to the constraints f(i) = 0 for i = 1..s. I construct a new function H = F + Σiλif(i) which has s terms added onto F to make H and the λi are unknown constants. Then H is a function of M+s variables. By fiat I then set Hi = 0 for i = 1 to M+s. What is the justification for doing that? Yes, the last 6 equations replicate the constraints. The first M equations combined are a statement that the rows of the R matrix are linearly dependent. I know that this is associated with a reduced R matrix rank. Then my Theorem 1 says this is a stationary point r. So my motivation for setting Hi = 0 is that it replicates the constraints AND it creates the condition that the R matrix rows are linearly dependent which I know means that r is a stationary point for F(r).
Am I somehow claiming that H is a function of M+s independent variables? If that were the case, then my conditions Hi = 0 would identify a stationary point of the function H. That IS what I am doing (setting all M+s Hi = 0) so it is AS IF I were trying to extremize the function H.