I also wondered how Leonhard Euler arrived at all those formulae and equations. In many cases he did a finite difference on the problem and then let h -> 0. Clever. Euler-Lagrange QuoteIn solving optimisation problems in function spaces, Euler made extensive use of this `methodof finite differences'. By replacing smooth curves by polygonal lines, he reduced the problem offinding extrema of a function to the problem of finding extrema of a function of n variables, andthen he obtained exact solutions by passing to the limit as n ! 1. In this sense, functions canbe regarded as `functions of infinitely many variables' (that is, the infinitely many values of x(t)at different points), and the calculus of variations can be regarded as the corresponding analog ofdifferential calculus of functions of n real variables.
Last edited by Cuchulainn
on February 11th, 2015, 11:00 pm, edited 1 time in total.