How did Lagrange discover Lagrange multipliers? Also, was it related to his work on the calculus of variations? And how did he originally understand/implement the technique?
[Math] History of Lagrange Multipliers
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math-historyreference-request
How did Lagrange discover Lagrange multipliers? Also, was it related to his work on the calculus of variations? And how did he originally understand/implement the technique?
Best Answer
Here is some information from the book A History of Analysis written by Hans Niels Jahnke
Lagrange presented his new conception in a memoir (Lagrange 1764), written to answer a question advanced in 1762 by the Academy of Paris, concerning the libration of the moon. More than fifteen years later (Lagrange 1780), he outlined a more elaborate and general version of the same program. He finally realised this program in the Méchanique analitique (Lagrange 1788).
Lagrange's idea has two clear sources. The first is the Bernoullian principle of virtual velocities for the equilibrium of any system of bodies animated by central forces. ... The second source is d'Alembert's principle for dynamics. ... Lagrange showed ((Lagrange $1780$, $218$-$220$) and (Lagrange $1788$, $216$-$227$)) that, if the forces are conservative, then $(5.7)$ gives rise to the equation
$$\sum_{i=1}^{\nu}\left[d\frac{\delta T}{\delta d\varphi_i}-\frac{\delta T}{\delta \varphi_i}+\frac{\delta U}{\delta \varphi_i}\right]\delta \varphi_i = 0\qquad(5.8)$$
where $T$ and $U$ are what we would now call the kinetic and the potential energy of the system. This is the original version of the so-called Lagrangian equation of motion. Its deduction from $(5.7)$ depends on the choice of the variable $\varphi_i$. We must ensure that the variations for these variables are mutually independent.
The Lagrangian equation of motion and the method of Lagrange multipliers are important contributions to the theoretical mechanics. However, they appear in both the $1780$ memoir on the libration of the moon and in the Méchanique analitique as tools to make possible an application of the quasi-algebraic method of undetermined coefficients to the general equation expressing the principle of virtual velocities, in order to deduce from it the equations of motion for any system of bodies.
The following text is essentially from P. Bussottis paper: On the Genesis of the Lagrange Multipliers
He introduced this mathematical approach in the framework of statics in order to determine the general equations of equilibrium for problems with constraints. At the beginning of his Mécanique Analytique (Ref $1$), Lagrange tackles statics and poses three principles as the foundations for the subject: (i) the principle of the lever (ii) the principle of the composition of forces and (iii) the principle of virtual velocities. It was this third principle virtual velocities where the Lagrange Multiplier occurred the first time.
So, if we consider a material point to which only two forces $P$ and $Q$ are applied and if $\frac{dp}{dt}$ and $\frac{dq}{dt}$ ($p$ and $q$ being the directions, respectively, of $P$ and $Q$) are the virtual velocities that the body receives, respectively, by the forces $P$ and $Q$, then the body is in equilibrium if $$P\frac{dp}{dt}+Q\frac{dq}{dt}=0$$ This equation represents the principle of the virtual velocities. The quantity $\frac{dp}{dt}$ is the moment of the force $P$.
At the beginning of Section $2$ (Ref. $1$, paragraph $1$), when the real mathematical treatment begins, Lagrange replaces $\frac{dp}{dt}$ with $dp$. He justifies the substitution in this way (Ref. $1$, page $24$):
So, the general equation of the equilibrium of a point on which three forces are applied will be
$$Pdp+Qdq+Rdr=0$$
In this way, Lagrange is able to solve the problem in a more general and in a simpler way. Particularly, he studies in detail the equilibrium under constraints. He examines, in a first instance, the equilibrium of a material point or of a system of material points that are submitted to constraints. If the constraining equations are $L=0, M=0$, . . . and if $L, M$ are functions of several variables, it follows necessarily that $dL=0, dM=0$, etc. Consequently, if $\lambda,\mu$ are arbitrary real numbers [Lagrange calls them quantités indéterminées or coefficientes indéterminées (Ref. $1$, page $70$)], we have $\lambda dL=0, \mu dM=0$, etc. He explains:
The general form of this differential equation is
$$P dp+Q dq+\cdots+\lambda dL+\mu dM\cdots = 0$$
Such equation is the general equation of equilibrium. Considering each variable of an orthogonal reference frame, we will have a particular equation of equilibrium. For example, in the direction $x$, we will have
$$P \frac{dp}{dx}+Q \frac{dq}{dx}+\cdots+\lambda \frac{dL}{dx}+\mu \frac{dM}{dx}\cdots = 0$$
From a physical point of view, the reaction of the constraint is equated to an agent force; indeed, $\lambda$ and $\mu$ are treated as forces and the expressions $\lambda dL$ and $\mu dM$ represent the virtual works that are realized by these forces (Lagrange continues to call them moments).
From a mathematical point of view, Lagrange points out that the difficulty consists in determining the values of $\lambda , \mu$, etc. Actually, this difficulty can be overcome easily (Ref. $1$, paragraph $4$, Section $4$, page $71$), since in fact the use of the constraints allows having a system where the number of the equations and the number of the variables is the same.
An example can be useful to interpret and clarify the Lagrange reasoning: if we have two agent forces $P$ and $Q$ and a constraint represented by the equation $L=0$, we will have (imagining it to be in the ordinary space) the three unknown quantities $x, y, z$ and the unknown quantity $\lambda$ , and we will have four equations
\begin{align*} P \frac{dp}{dx}+Q \frac{dq}{dx}+\lambda \frac{dL}{dx}=0\\ P \frac{dp}{dy}+Q \frac{dq}{dy}+\lambda \frac{dL}{dy}=0\\ P \frac{dp}{dz}+Q \frac{dq}{dz}+\lambda \frac{dL}{dz}=0\\ L=0 \end{align*}
If the multiplier is interpreted as a new coordinate, the last equation can be replaced by another one that must be valid in the case of equilibrium: \begin{align*} P \frac{dp}{d\lambda}+Q \frac{dq}{d\lambda}+ \lambda \frac{dL}{d\lambda}=0 \end{align*}