The calculus of Newton and Leibniz can find the highest point of a curve: it is where the derivative vanishes. But there are questions of another kind. What is the shortest path between two points? What shape should a slide have so that a ball runs down it fastest? What curve does a hanging chain make? In these questions the answer is not a number but a whole curve, chosen from infinitely many possible curves.

The brachistochrone

In 1696 Johann Bernoulli issued a challenge to the mathematicians of Europe. Given two points $A$ and $B$, with $B$ lower and not directly below $A$, along which curve will a ball, sliding without friction under its own weight, go from $A$ to $B$ in the least time? The name of the problem comes from Greek: brachistos, shortest, and chronos, time.

The straight line is the shortest path, but not the fastest. It pays to start by dropping steeply, to gain speed early, even if the path gets longer. Johann, his brother Jakob, Leibniz, l’Hôpital and an anonymous author whom everyone recognised as Newton answered the challenge. The answer is a cycloid, the curve traced by a point on a rolling wheel.

Each solution was a different clever trick. Johann, for instance, imagined the ball as a ray of light crossing layers of glass of decreasing density and used the law of refraction. What was missing was a general method.

Euler’s method

Euler spent years looking for that method and published it in 1744, in a book called A Method for Finding Curved Lines Enjoying Properties of Maximum or Minimum. The idea was geometric: replace the curve by a polygon with many sides, move a single vertex and see the effect on the quantity to be optimised. It worked, but the calculations were long and each problem required its own drawing.

The letter of 1755

On 12 August 1755, Euler received a letter from an unknown 19-year-old in Turin. Lagrange proposed something else: instead of moving one vertex, vary the whole curve at once. If $y(x)$ is the curve sought, he considered a neighbouring curve,

$$y(x)+\delta y(x),$$

where $\delta y$ is a small perturbation that vanishes at the ends, because the endpoints $A$ and $B$ are fixed. The $\delta$, a Greek d, was the new symbol: it denotes a variation of the function, just as Leibniz’s $d$ denotes a variation of the variable.

Many of these questions ask us to make an integral as small or as large as possible,

$$J=\int_a^b F(x,y,y')\,dx,$$

where $F$ depends on the point, on the height of the curve and on its slope. The length of a curve, for example, is

$$J=\int_a^b\sqrt{1+y'^2}\,dx.$$

The equation

Lagrange’s calculation fits in three steps. First, the variation of the integral is the sum of the variations caused in $y$ and in $y'$:

$$\delta J=\int_a^b\left(\frac{\partial F}{\partial y}\,\delta y+\frac{\partial F}{\partial y'}\,\delta y'\right)dx.$$

Second, since $\delta y'$ is the derivative of $\delta y$, an integration by parts moves the derivative onto the other factor. The boundary term disappears, because $\delta y$ is zero at the ends:

$$\delta J=\int_a^b\left(\frac{\partial F}{\partial y}-\frac{d}{dx}\frac{\partial F}{\partial y'}\right)\delta y\,dx.$$

Third, if the curve is the best one, $\delta J$ must be zero for any perturbation $\delta y$. That is only possible if the bracket is zero at every point:

$$\boxed{\ \frac{\partial F}{\partial y}-\frac{d}{dx}\frac{\partial F}{\partial y'}=0\ }$$

This is the Euler–Lagrange equation. Euler had already found it by way of polygons; Lagrange reached it without any drawing, by a calculation that works for any $F$. The problem of finding one curve among all curves had become the problem of solving a differential equation.

The shortest path

Let us test it. For length, $F=\sqrt{1+y'^2}$ does not depend on $y$, so $\frac{\partial F}{\partial y}=0$ and the equation says that

$$\begin{aligned}&\frac{d}{dx}\left(\frac{y'}{\sqrt{1+y'^2}}\right)=0\\ &\Longrightarrow\quad \frac{y'}{\sqrt{1+y'^2}}=\text{constant}.\end{aligned}$$

If that fraction is constant, so is $y'$: the slope never changes, and the curve is a straight line. Everyone knew it, but now it is a theorem that comes out of a calculation. With the same method, the brachistochrone gives the cycloid, the hanging chain gives the catenary, and the curve of fixed length enclosing the largest area gives the circle.

Your analytical solution of the isoperimetric problem contains, as far as I can see, everything one could wish for in this field, and I am extremely glad that this theory, with which I have been occupied almost alone since the first attempts, has been brought by you to the greatest perfection.

— Leonhard Euler, letter to Lagrange (2 October 1759)

In 1756 Euler presented to the Berlin Academy an account of Lagrange’s method and gave the new branch its name: calculus variationum, the calculus of variations. Euler, already the most famous mathematician in Europe, gave up part of the glory to the unknown young man. Lagrange never forgot it.

Exercise: which curve joins (0, 0) to (1, 2) making the integral of y′² as small as possible?

With $F=y'^2$, we have $\frac{\partial F}{\partial y}=0$ and $\frac{\partial F}{\partial y'}=2y'$. The equation gives $\frac{d}{dx}(2y')=0$, that is, $y''=0$: the curve is a straight line $y=cx+d$. The endpoints give $d=0$ and $c=2$, and the answer is $y=2x$. The minimum value is $\int_0^1 2^2\,dx=4$.

Exercise: what curves come from F = y′² − y² ?

Now $\frac{\partial F}{\partial y}=-2y$ and $\frac{\partial F}{\partial y'}=2y'$. The equation becomes $-2y-2y''=0$, that is,

$$y''=-y.$$

The solutions are $y=A\cos x+B\sin x$. It is the equation of a spring. This is no coincidence: $y'^2$ plays the role of the energy of motion and $y^2$ that of the spring’s energy. It is the starting point of the next text.

Sources: biography of Lagrange, MacTutor · Herman H. Goldstine, A History of the Calculus of Variations from the 17th through the 19th Century (1980) · Craig Fraser, “J. L. Lagrange’s Early Contributions to the Principles and Methods of Mechanics”, Archive for History of Exact Sciences (1983).

All Lagrange topics · ← The life of Joseph-Louis Lagrange · The Analytical Mechanics: a book without figures →

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