IV. On the discrimination of maxima and minima solutions in the calculus of variations

Abstract

The criteria for distinguishing between the maximum and minimum values of integrals have been investigated by many eminent mathematicians. In 1786 Legendre gave an imperfect discussion for the case where the function to be made a maximum is ʃ f (x,y, dy / dx ) dx . Nothing further seems to have been done till 1797, when Lagrange pointed out, in his ‘Théorie des Fonctions Analytiques,' published in 1797, that Legendre had supplied no means of showing th at the operations required for his process were not invalid through some of the multipliers becoming zero or infinite, and he gives an example to show that Legendre’s criterion, though necessary, was not sufficient. In 1806 Brunacci, an Italian mathematician, gave an investigation which has the important advantage of being short, easily compiehensible, and perfectly general in character, but which is open to the same objection as that brought against Legendre’s method. The next advance was made in 1836 by the illustrious Jacobi, who treats only of functions containing one dependent and one independent variable. Jacobi says (Todhunter, Art. 219, p. 243): “I have succeeded in supplying a great deficiency in the Calculus of Variations. In problems on maxima and minima which depend on this calculus no general rule is known for deciding whether a solution really gives a maximum or a minimum, or neither. It has, indeed, been shown that the question amounts to determining whether the integrals of a certain system of differential equations remain finite throughout the limits of the integral which is to have a maximum or a minimum value. But the integrals of these differential equations were not known, nor had any other method been discovered for ascertaining whether they remain finite throughout the required interval. I have, however, discovered that these integrals can be immediately obtained when We have integrated the differential equations which must be satisfied in order that the first variation may vanish.” Jacobi then proceeds to state the result of his transformation for the cases where the function to be integrated contains x, y, dy / dx , and x, y, dy / dx 2 , and in this solution the analysis appears free from all objection, though, where he proceeds to consider the general case, the investigation does not appear to be quite satisfactory in form, inasmuch as higher and higher differential coefficients of By are successively introduced into the discussion (see Art. 5). Jacobi’s analysis is much more complicated than Brunacci's, its advantage being that the coefficients used in the transformation could be easily determined; hence it supplied the means of ascertaining whether they became infinite or not.