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\begin{document}

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\section*{Lagrange multiplier}

\hypertarget{contents}{}\section*{{Contents}}\label{contents}

\noindent\hyperlink{idea_following_loomissternberg}{Idea following Loomis-Sternberg}\dotfill \pageref*{idea_following_loomissternberg} \linebreak
\noindent\hyperlink{applications}{Applications}\dotfill \pageref*{applications} \linebreak
\noindent\hyperlink{ApplicationToSpectralTheory}{To spectral theory}\dotfill \pageref*{ApplicationToSpectralTheory} \linebreak
\noindent\hyperlink{related_concepts}{Related concepts}\dotfill \pageref*{related_concepts} \linebreak
\noindent\hyperlink{references}{References}\dotfill \pageref*{references} \linebreak


\hypertarget{idea_following_loomissternberg}{}\subsection*{{Idea following Loomis-Sternberg}}\label{idea_following_loomissternberg}

$X$ -- finite dimensional real vector space

$U\subset X$ open

$F: U\to \mathbf{R}$ differentiable function

$S\subset U$ a smooth submanifold, which can be represented as a zero set of a differentiable map $G: U\to Y$, whre $Y$ is a real vector space and such that $d G_x$ is surjective for each $x\in S$.

We want to minimize $F(x)$ for $x\in S$. It won't work to set $d F_x = 0$ and solving for $x$ as $x$ will not be a critical point of $F$ in general. The Lagrange multipliers are used to define another function $L$ such that solving $d L_x = 0$ gives extrema of the constrained extremization problem.

\textbf{Theorem} (Loomis-Sternberg 3.12.2) Suppose $F$ has a maximum on $S$ at $x$. Then there is a function(al) $l$ in $Y^*$ such that $x$ is a critical point of the function $F - l\circ G$.

The proof uses implicit function theorem and the usual extremization arguments.

To get to a more familiar form of Lagrange multipliers, one uses the local coordinates $(x_1,\ldots,x_n)$ on $U$ and sets $Y = \mathbf{R}^m$, so that $G = (g^1,\ldots, g^n)$. Now $l: Y\to\mathbf{R}$ will be of the form $l(y_1,\ldots,y_m) = \sum_{i = 1}^m \lambda_i y_i$ and $F - l\circ G = F - \sum_{i = 1}^m \lambda_i g^i$ and $d (F - l\circ G) = 0$ gives

\begin{displaymath}
\frac{\partial F}{\partial x_j} - \sum_{i = 1}^m\lambda_i\frac{\partial g^i}{\partial x_j} = 0,\,\,\,\,\,\,\,\,j = 1,\ldots, n.
\end{displaymath}
This is $n$ equations, which together with $m$ equations $G = (g^1,\ldots, g^n) = 0$ for $S$ give $m+n$ equations for $m+n$ unknowns $x_1,\ldots, x_n, \lambda_1,\ldots,\lambda_m$. The last $m$ variables here are the Lagrange multipliers.

\hypertarget{applications}{}\subsection*{{Applications}}\label{applications}

\hypertarget{ApplicationToSpectralTheory}{}\subsubsection*{{To spectral theory}}\label{ApplicationToSpectralTheory}

The method of Lagrange multipliers affords an elementary proof of the [[spectral theorem]] for finite-dimensional real [[vector spaces]], one which does not involve passage to the [[complex numbers]] and the [[fundamental theorem of algebra]].

\begin{uprop}
Let $A$ be a real symmetric $n \times n$ matrix. Then $A$ is diagonalizable over the real numbers.

\end{uprop}
\begin{proof}
Consider the problem of maximizing the function $f(x) = \langle x \vert A \vert x \rangle$ where $x \in \mathbb{R}^n$ is subject to the constraint $\langle x \vert x \rangle = 1$. (Such an extreme point exists, say by compactness.) By the symmetry of $A$, the [[gradient]] of $f$ is easily calculated to be $\nabla f (x) = 2 A x$, whereas the gradient of the [[Euclidean norm]] $\langle x \vert x \rangle$ is $2 x$. At a point $x$ where a maximum is attained, we have $\nabla f(x) = 2 A x = \lambda (2 x)$ for some Lagrange multiplier $\lambda$. Thus $x$ is an eigenvector of $A$ with eigenvalue $\lambda$. The usual arguments show that $A$ restricts to a [[self-adjoint operator]] on the hyperplane orthogonal to $x$; by picking an orthonormal basis of this hyperplane, we may represent this restriction of $A$ by a real symmetric matrix of size $(n-1) \times (n-1)$, and the argument repeats.

\end{proof}
\hypertarget{related_concepts}{}\subsection*{{Related concepts}}\label{related_concepts}

\begin{itemize}%
\item [[auxiliary field]]


\item [[gauge fixing]]



\end{itemize}
\hypertarget{references}{}\subsection*{{References}}\label{references}

\begin{itemize}%
\item wikipedia \href{http://en.wikipedia.org/wiki/Lagrange_multiplier}{Lagrange multiplier}
\item Springer eom: \href{http://eom.springer.de/L/l057190.htm}{Lagrange multipliers}, \href{http://eom.springer.de/p/p073780.htm}{Pontrjagin maximum principle}
\item Lynn H. Loomis, [[S. Sternberg]], \emph{Advanced calculus}, section 3.12 ``Submanifolds and Lagrange multipliers'' [[LoomisSternberg13s2.djvu:file]]
\item Robert Hermann, \emph{Some differential-geometric aspects of the Lagrange variational problem}, Illinois J. Math. \textbf{6}, 1962, 634--673, \href{http://www.ams.org/mathscinet-getitem?mr=145457}{MR145457},\href{http://projecteuclid.org/euclid.ijm/1255632711}{euclid}
\item Juan Carlos Marrero, David Mart\'i{}n de Diego, Ari Stern, \emph{Lagrangian submanifolds and discrete constrained mechanics}, slides, \href{http://ccom.ucsd.edu/~astern/pdfs/dspde20100601.pdf}{pdf}
\item Manuel de Le\'o{}n, David Mart\'i{}n Diego, \emph{Solving non-holonomic Lagrangian dynamics in terms of almost product structures}, Extracta Math. 11 (1996), no. 2, 325--347, \href{http://digital.csic.es/bitstream/10261/2232/1/solving.pdf}{pdf} \href{http://www.ams.org/mathscinet-getitem?mr=1437457}{MR97m:58079}

\end{itemize}
[[!redirects Lagrange multipliers]]



\end{document}