Who was responsible for finding sufficient conditions for functional extrema?

Question

In the calculus of variations, there is a well-known sufficient condition for weak functional extrema, involving conjugate points and the strengthened Legendre condition ($f_{y'y'} > 0$). Who was responsible for this discovery? Charles Fox (An Introduction to the Calculus of Variations, p. 42) credits Jacobi (1837), whereas Morris Kline (Mathematical Thought From Ancient to Modern Times, Vol. 2, p. 748) credits Weierstrass (1879). This is no small discrepancy of forty-two years. Can anyone give an authoritative answer on who is right? References would be good.

Answer 1

Jacobi thought that the Euler condition, strengthened Legendre condition, and absence of conjugate points on $(a,b]$ were together sufficient conditions for $y$ to be a minimum of the integral [1,2] $$ I = \int_a^b F(y,y^\prime,x) dx. $$ However, Weierstrass proved that this is not true, that the given conditions are sufficient only for weak variations, and that the given three together with a fourth gave sufficient conditions also for strong variations [1,2,3].

So, who you want to credit the theorem to is somewhat up to you. Jacobi came first, but stated the theorem without a complete proof and without the concept of weak VS strong variations. Weierstrass rigorously proved both cases.

The above story is actually laid out pretty well in the book by Kline that you referenced: "Jacobi also concluded that an extremal (a solution of Euler's equation) taken between A and B for which $f_{y^\prime y^\prime} > 0$ along the curve and for which no conjugate point exists between $A$ and $B$ (or at $B$) furnishes a minimum for the original integral... Actually, these sufficient conditions were not correct... Weierstrass did prove in 1879 that for weak variations the three conditions ... are indeed sufficient conditions" [1] (emphasis on weak variations added by me).

Here are some additional references corroborating the story:

"Jacobi's condition was originally erroneously stated as a sufficient condition, and was shown by Weierstrass to be insufficient for strong minima." [2] (See pp. 18.)

"In his earlier work, prior to the year 1879, he [Weierstrass]... proved for the first time that the condition $\delta^2 I > 0$ is sufficient for the existence of a weak minimum." [3]

Note also that any of the references crediting Jacobi with the theorem for the weak minimum don't provide these historical details [4, 5]. It seems to me that some authors simply credit the correct part of Jacobi's statement to him, whereas other authors give the credit mainly to Weierstrass, who corrected and extended Jacobi's statement, and gave rigorous proof.

[1] M. Kline. Mathematical Thought From Ancient to Modern Times, Vol. 2, p. 747--748.

[2] E. R. Hedrick. On the Sufficient Conditions in the Calculus of Variations. Bull. Amer. Math. Soc. 9, 11-24 (1902).

[3] M. Giaquinta and S. Hildebrandt. Calculus of Variations I: The Lagrangian Formalism. Springer-Verlag Berlin (2004). ISBN 3-540-50625-X. pp. 261.

[4] C. Fox. An Introduction to the Calculus of Variations, p. 42.

[5] Sufficiency and the Jacobi Condition in the Calculus of Variations. F. H. Clarke and V. Zeidan. Can. J. Math. 38 (5), 1199-1209 (1986).