Why is the integrated difference of the kinetic and potential energies the quantity to be minimized in Hamilton’s principle? I use simple arguments to convert the problem of finding the path of a particle connecting two points to that of finding the minimum potential energy of a string. The mapping implies that the configuration of a nonstretchable string of variable tension corresponds to the spatial path dictated by the principle of least action; that of a stretchable string in space–time is the one dictated by Hamilton’s principle. This correspondence provides the answer to the question.
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Strictly speaking, extremal (maxima or minima) are a subset of stationary paths (or states). However, the term principle of least action is more frequently used than the more precise principle of stationary action.
The title of this section was taken from Ref. 10, p. 209.
It could be objected that Hamilton did not know the wave equation for light, but the argument applies to any scalar wave in a nonhomogeneous medium. For example, Eq. (20) could represent a sound wave in a medium with varying density.
