The History of Mathematics: A Brief Course

3. BRANCHES AND ROOTS OF THE CALCULUS 481

extrema into absolute (unconstrained) extrema. Euler had given an explanation of

this process earlier. Woodhouse (1810, p. 79) thought that Lagrange's systemati-

zation actually deprived Euler's ideas of their simplicity.

Second-variation tests for maxima and minima. Like the equation f'(x) = 0 in

calculus, the Euler equation is only a necessary condition for an extremal, not

sufficient, and it does not distinguish between maximum, minimum, and neither.

In general, however, if Euler's equation has only one solution, and there is good

reason to believe that a maximum or minimum exists, the solution of the Euler

equation provides a basis to proceed in practice. Still, mathematicians were bound

to explore the question of distinguishing maxima from minima. Such investigations

were undertaken by Lagrange and Legendre in the late eighteenth century.

In 1786 Legendre was able to show that a sufficient condition for a minimum

of the integral

i(y) = / f(x,y,y')dx,

J a

at a function satisfying Euler's necessary condition, was > 0 for all ÷ and that

a sufficient condition for a maximum was -^ö < 0.

In 1797 Lagrange published a comprehensive treatise on the calculus, in which

he objected to some of Legendre's reasoning, noting that it assumed that certain

functions remained finite on the interval of integration (Dorofeeva, 1998, p. 209).^9

Jacobi: sufficiency criteria. The second-variation test is strong enough to show that
a solution of the Euler equation really is an extremal among the smooth functions
that are "nearby" in the sense that their values are close to those of the solution
and their derivatives also take values close to those of the derivative of the solution.
Such an extremal was called a weak extremal by Adolf Kneser (1862-1930). Jacobi
had the idea of replacing the curve y(x) that satisfied Euler's equation with a
family of such curves depending on parameters (two in the case we have been
considering) y(x, ct\, a 2 ) and replacing the nearby curves y + Sy and y' + 5y' with
values corresponding to different parameters. In 1837—see Dorofeeva (1998) or
Fraser (1993)—he finally solved the problem of finding sufficient conditions for an
extremal. He included his solution in the lectures on dynamics that he gave in
1842, which were published in 1866, after his death. The complication that had
held up Jacobi and others was the fact that sometimes the extremals with given
endpoints are not unique. The most obvious example is the case of great circles on
the sphere, which satisfy the Euler equations for the integral that gives arc length
subject to fixed endpoints. If the endpoints happen to be antipodal points, all great
circles passing through the two points have the same length. Weierstrass was later
to call such pairs of points conjugate points. Jacobi gave a differential equation
whose solutions had zeros at these points and showed that Legendre's criterion was
correct, provided that the interval (a, b] contained no points conjugate to o.

Weierstrass and his school. A number of important advances in the calculus of

variations were due to Karl Weierstrass, such as the elimination of some of the more

(^9) More than that was wrong, however, since great circles on a sphere satisfy Legendre's criteria,
but do not give a minimum distance between their endpoints if they are more than 180° long.