The Early Period of the Calculus of Variations by Paolo Freguglia & Mariano Giaquinta

Author:Paolo Freguglia & Mariano Giaquinta
Language: eng
Format: epub
Publisher: Springer International Publishing, Cham

He observes that with these substitutions all differentials of y of whatever order will be removed and besides dx no other differential will remain; for instance6

Moreover, if

denotes the arc length, then

while ,

Next, Euler discusses the quantity to be minimized or maximized, that he calls the Maximum or minimum formula, concluding that it must be referred to the same abscissa the curves are referred to and will be any function W depending not only on x and y but also on quantities which in turn depend on them, such as etc. (called the determinate functions) and on integral formulas arising from these, as for instance the path length s (in contraposition, called the indeterminate functions). Euler concludes that the formula must have the form7

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