I'm interested in a historical perspective on pde.
I would like to know more about the original derivation of pde. It seems like d'Alembert was working on the one dimensional wave equation $$ \partial_{tt} u = \partial_{xx}u $$ which was subsequently generalized to multiple dimension by Euler and Bernoulli leading to $$ \partial_{tt} u = \Delta u $$ Only later Laplace was studying the equation named after him in the context of gravitational potential fields. I got those informations from the very nice article "Partial Differential Equations in the $20$th Century" by Brezis and Browder. However, they are not referencing the original works and say nothing about the way in which those equations came up. Can someone give references to those early works which are describing what is their motivation for studying these equations? Or maybe someone can summarize the ideas that lead to these equations. In particular I want to know if the arguments were based on limits of pointwise properties or more on energy minimization principles. I will elaborate my reason for asking this:
The two interpretations of the Laplace operator that I am aware of are either the pointwise description of it as a certain rescaled limit of the average difference between the value at this point and the values on small spheres around it, or the viewpoint of "energy" minimization, e.g. harmonic functions have minimal $L^2$ norm under prescribed boundary values. One could argue that the widely used weak formulation of pde is taking a certain middle ground between those views. It generalizes the pointwise description but it also doesn't need the energy minimization itself (by which I mean that we can also deal with nonsymmetric problems, which as far as I know, can not be characterized as energy minimizers), but rather the first order optimality conditions of the energy functional (or something that looks similiar in case there is no energy functional). The usual way to introduce the weak formulation is to start with the pointwise equation and derive the weak formulation by Gauss' theorem or something similiar. However, I was under the impression that energy minimization is a very common physical principle so I am wondering whether the historical development did not actually look like this: By some physical argument, the solution describing our physical process is minimizing a certain kind of energy. Hence (ignoring regularity) it satisfies this pde. Now to show existence of a solution to a pde, we proceed as follows: By some mathematical argument, the solution of the pde is minimizing a certain kind of energy and the theory of Sobolev Spaces starts. In this exaggerated view it seems like the pointwise description is an artificially introduced unnecessity. I am not arguing against the pointwise description (which I think is important in e.g. analytic function theory as started by Riemann and probably for a whole lot more of reasons that I am ignorant of) but I wonder how much truth there is in this description of the history.
Maybe someone even knows of earlier work dealing with pde?