VARIATIONS

VARIATIONS, CALCULUS OF. 1. The object of this calculus is the discovery of the form of a function which shall fulfil certain conditions, not expressed in direct terms, but involved in a finite integral. The first instance of a problem of this kind appears in a scholium to Prop. 34, b. ii. of the Principia, published in 1687. (See Prob. 2. below.) Newton gave the correct result; but he supplied no demonstration, nor do we find any method of solution applicable to such cases until ten years afterwards. In the Acta Eruditorum for 1696, John Bernoulli enunciated the problem of the brachystochrone, (Prob. 1. below,) and invited mathematicians to give a solution. After a considerable interval, his brother, James Bernoulli, gave in his result, which was, that the curve required is the cycloid. As the mode of demonstration which he adopted is very nearly that which was employed by all the earlier writers on the subject, it will not be uninteresting to exhibit it here.

2. The problem is the following: "To find that curve down which a body falling by the force of gravity will move from A to B in the shortest possible time."

The principles employed in the solution are these two:

1st, that when a quantity is a maximum or minimum, a slight change in the variables will produce no variation in the value of the function; 2d, that what is true of the whole quantity is likewise true of every portion of it.

The second principle requires that the time down PQR should be less than the time down any other line, as PSR;

Variations, and by applying the first principle to this property, we deduce the fundamental equation,

time down PQR = time down PSR ultimately.

Hence,

a well-known property of the cycloid.

3. Of the two principles employed in this solution, the first is equivalent to the theorem demonstrated in FLUXIONS, art. 62, and is the basis of all solutions. The second, although in the case in question it is actually true, is an assumption not warranted by the nature of the problem. It does not appear that any geometrical solution is exempt from this objection, and we cannot in consequence date the existence of the Calculus of Variations earlier than 1744, when Euler published his treatise, entitled Methodus inveniendo Lineas Curvas proprietate maximi minimi gaudentes. This work was followed by a memoir from the pen of Lagrange, published in the second volume of the Miscellanea Taurinensia. By the introduction of the symbol $\backslash delta$ to express that change which is termed a variation, Lagrange may be said to have perfected the calculus; for although certain extensions were afterwards made by himself and others, no change was afterwards introduced into the general process. We shall therefore conclude our brief sketch at this point, and refer the reader for further information to the following works: the Acta Eruditorum for 1696 and the following years; the collected works of James and John Bernoulli; Memoires de l'Acad. des Sci. 1706, 1718, &c.; Brook Taylor's Methodus Incrementorum; the Petersburg Commentaries, vols. vi. and viii.; which, together with Euler's tract, quoted above, and his last memoir in the New Petersburg Com. vol. x. contain Euler's writings on the subject: the Turin Miscellanies, vols. ii. and iv. and the Theorie des Fonctions Analytiques, contain Lagrange's perfecting of the calculus. The reader will find in Woodhouse's Isoperimetrical Problems a complete history of the calculus, as well as an admirable digest of the different methods employed.

4. We proceed to the investigation of the theorem which is the basis of the calculus of variations.

To find the change or variation of a formula comprehended under the integral sign, when the variables $x$ and $y$, on which it depends, receive the increments $\backslash delta\; x$ and $\backslash delta\; y$.

Let $u\; =\; \backslash int\; \backslash gamma\; dx$ be the formula under consideration; in

which $\backslash gamma$ involves $x$, $y$, $\backslash frac\{dy\}\{dx\}$, $\backslash frac\{d^2y\}\{dx^2\}$ &c.

Denote $\backslash frac\{dy\}\{dx\}$ by $p$, $\backslash frac\{d^2y\}\{dx^2\}$ by $q$, &c.; the partial differential coefficients $\backslash frac\{d\backslash gamma\}\{dx\}$ by $M$, $\backslash frac\{d\backslash gamma\}\{dy\}$ by $N$, $\backslash frac\{d\backslash gamma\}\{dp\}$ by $P$, &c.; the increment of $\backslash int\; \backslash gamma\; dx$ by $\backslash delta\; \backslash int\; \backslash gamma\; dx$, &c.

Also if $\backslash delta\; y\; -\; p\; \backslash delta\; x$ be denoted by $\backslash omega$, we have

hence putting $p$ for $y$ and $\backslash therefore\; q$ for $p$, we obtain

By substitution,

The last line of this very elegant expression is due to Euler; the rest of the formula to Lagrange.

Let $x\_1,\; x\_2,\; y\_1,\; y\_2$ be the values of $x$ and $y$ at the two limits; then the whole value of $\backslash delta\; u$ between the limits is,

5. To satisfy the conditions of a maximum or minimum value of $u$, the value of $\backslash delta\; u$, expressed only as far as to the first powers of $\backslash delta\; x,\; \backslash delta\; y,\; \backslash delta\; c$, must be equal to zero.

But this quantity consists of two parts; the one an integrated expression, depending only on the values of $\backslash delta\; x,\; \backslash delta\; y,\; \backslash delta\; c$; the other an unintegrated expression, depending on the general values of $\backslash delta\; x,\; \backslash delta\; y$. Now it is evidently possible to make the latter expression assume an infinity of different values, corresponding to an assigned value of the former. It is therefore impossible to render the whole expression equal to zero for all values of $\backslash omega$, without making the two parts separately zero.

Our conditions are therefore,

6. It may be remarked, that the object of our investigation is to discover a mode of satisfying certain conditions by means of establishing a relation between $y$ and $x$. To effect this, we seek to separate these quantities from the quantities $\backslash delta\; x$ and $\backslash delta\; y$. This will explain why the unintegrated part of the expression has been reduced to the form in which we left it. Had $\backslash omega$ been implicated with $x$ and $y$,

relation would have been assigned, not between $x$ and $y$, but between $x$, $y$, and $\backslash omega$. We may add, that if $\backslash int\; y\; dx$ be an integrable expression, the equation (1) is an identical equation.

In the above investigation it has been assumed that the other quantities which enter into the function $\backslash gamma$, together with $x$ and $y$, are absolute constants. If this hypothesis be not correct, our formulae require to be modified. Suppose $a$, $b$, &c., to enter the function $\backslash gamma$.

Let $\backslash delta\backslash gamma\; =\; M\backslash delta\; x\; +\; N\backslash delta\; y\; +\; P\backslash delta\; p\; +\; \backslash delta\; c\; +\; A\backslash delta\; a\; +\; \backslash delta\; c\; \backslash dots$ the addition to be made to the expression for $\backslash delta\backslash omega$ is $\backslash int\; (\backslash delta\; a\; +\; \backslash dots)\; dx$. But $\backslash delta\; a$ &c. depend not on the general variation, but only on the variation at the limits, (see Prob. 4); here they are independent of $\backslash delta\; x$ and $\backslash delta\; y$, and consequently the part of equation (2), which thus becomes

Had our object been merely to determine the form of a function which should satisfy certain conditions, we might have rested content with causing one of the quantities, as $x$ only, to vary. By this means equation (1) would have been obtained. There are however many problems which depend on the relation between $\backslash delta\; x$ and $\backslash delta\; y$, at least at the limits, (see Prob. 3.) At the commencement of the variation, for instance, we cannot remove to any point we please, from the circumstance that we are compelled to begin in a certain manner. Hence equation (2) is a most important part of the result; and by assigning a relation between the things given and sought at the extremities, shews us the portion of the extremities of the curve, and thereby restricts the solution from a kind of curve (as a cycloid) to a specific curve, (a cycloid of known dimensions and position).

From equation (1) we might easily deduce a number of expressions, for particular cases, easy of application. Thus, if $\backslash gamma$ contains only $y$ and $p$, the formula (1) gives

As the reader will readily supply such formulae for himself, we proceed to exhibit the solution of a few problems.

Prob. 1. To find the curve of quickest descent from one given point to another.

Let $x$ and $y$ be the co-ordinates of a point in the curve, $y$ being measured vertically downwards from the upper point.

corresponding to $\backslash gamma$ contains only $y$ and $p$.

which is the differential equation to a cycloid. We may solve the problem by taking $x$ as the vertical co-ordinate, in which case $\backslash gamma$ will contain only $p$, and equation (1) will

the same result as before, with the exception that $x$ and $y$ are interchanged. The solution of this equation shews that the cusp of the cycloid is at the highest point, and its axis vertical.

Prob. 2. Required the curve which by revolution round its axis generates the solid on which the resistance produced by motion in a fluid shall be less than on any other curve having the same extreme co-ordinates.

This equation expresses the same property as that which Newton, without demonstration, assigned to the curve.

Prob. 3. To find the shortest distance between two given curves.

The expression for $\backslash gamma$ is here $\backslash sqrt\{1+p^2\}$ where, as the line on which the distance is measured is a right line, $p=c$. But by formula (2)

$\backslash gamma\_1\; \backslash delta\; x\_1\; -\; \backslash gamma\_0\; \backslash delta\; x\_0\; +\; P\_1\; \backslash omega\_1\; -\; P\_0\; \backslash omega\_0\; =\; 0$, when $\backslash delta\; x\_1$, &c. are indefinitely small. Also, if the curves have no connexion, we may make $\backslash delta\; x\_1\; \backslash delta\; y\_1$ assume what values soever we please, without affecting $\backslash delta\; x\_0\; \backslash delta\; y\_0$; we must therefore have separately

But, since the measurement must begin in the first curve, the limit of $\backslash frac\{\backslash delta\; y\_0\}\{\backslash delta\; x\_0\}$ expresses the tangent of the angle in

which the tangent to the first curve cuts the axis. Our equation, then, shews that the line of shortest distance cuts the first curve at right angles. For the same reason it cuts the second curve also at right angles. Its position is consequently determined.

Prob. 4. To find the curve of quickest descent from one given curve to another.

Let $h$ be the vertical ordinate of the point at which motion begins:

it is evident that the curve is a cycloid. To determine the angles at which the cycloid cuts the given curves, we have the following equation, (3): $\backslash gamma\_1\; \backslash delta\; x\_1\; -\; \backslash gamma\_0\; \backslash delta\; x\_0\; +\; P\_1\; \backslash omega\_1\; -\; P\_0\; \backslash omega\_0\; +$

1. If the body starts from a given point in the first curve, that curve performs no part in the problem. Here $\backslash delta\; x\_0$, $\backslash omega\_0$, and $\backslash delta\; h$, are all zero.

point in which the cycloid cuts it, is equal to $-\backslash frac\{1\}\{p\_1\}$: which shews that the curve and cycloid cut each other at right angles.

2. If the motion is supposed to commence at the same horizontal line, whatever be the point at which the body reaches the first curve; we have $h=d$ constant, and the equation gives

$\backslash gamma\_1\; \backslash delta\; x\_1\; -\; \backslash gamma\_2\; \backslash delta\; x\_2\; +\; P\_1\; \backslash omega\_1\; -\; P\_2\; \backslash omega\_2\; =\; 0$, from which it is evident, as in Problem 3, that the cycloid cuts both curves at right angles.

3. If motion begins at the first curve, $h=y\_0$ and $\backslash frac\{dy\}\{dh\}\; =\; -\backslash frac\{dy\_0\}\{dy\}\; =\; -N\; =\; -\backslash frac\{dP\}\{dx\}\; \backslash left(\; \backslash because\; N\; =\; \backslash frac\{dP\}\{dx\}\; =\; 0\; \backslash right)$; hence our equation becomes

From the first of these it follows that the lower curve is cut at right angles. The second gives

Also the first equation is $c\; +\; P\_1\; \backslash frac\{\backslash delta\; y\_0\}\{\backslash delta\; x\_1\}\; =\; 0$.

$\backslash therefore\; \backslash frac\{\backslash delta\; y\_0\}\{\backslash delta\; x\_2\}\; =\; \backslash frac\{\backslash delta\; y\_0\}\{\backslash delta\; x\_1\}$; from which it appears that the tangents to the two curves at the two points of section are parallel to one another.

10. We have hitherto solved only problems of absolute maxima and minima. But a very simple consideration will enable us to apply the same formulae to the investigation of relative maxima and minima. The problem of isoperimetricals (prob. 5 below) will illustrate this class of questions. Here the integrated function is not required to be a maximum or a minimum absolutely, but only one consistent with the further condition that another integrated function shall not change its value. Although, therefore, $\backslash delta\; f\_1\; dx$ is actually zero, yet $\backslash delta\; x$ and $\backslash delta\; y$ are not, as in our previous investigations, any quantities whatever, but only such as will consist with the required condition that $\backslash delta\; f\; \backslash sqrt\{1+p^2\}\; dx$ shall also be zero. To Euler we owe the idea of substituting for the quantity $u$, which is made to vary, not $f\_1\; dx$, as in other cases, but $f\_1\; dx\; +\; a\; f\_2\; dx$; $f\_2\; dx$ being the quantity which is to remain unchanged. By this substitution we determine the relation between $x$ and $y$, which renders $\backslash gamma\; +\; a\; v$ a maximum or a minimum. But as that relation involves the arbitrary constant $a$, we restrict it by assigning to that quantity such a value as shall render $f\_2\; dx$ of the required magnitude. We have therefore found a relation, which not only makes the sum of two quantities a maximum or a minimum, but which likewise reduces one of those quantities to a specified value. This relation then evidently makes the other quantity a maximum or a minimum. Thus the problem is solved in all its generality.

Prob. 5. AB is a given line, PMQ a line perpendicular

to AB at the point M. The points P and Q are supposed to generate two curves by the motion of the line perpendicular to AB, having a relation such, that although both are unknown, PM is a known function of QM. It is required to determine both curves, when the length of AQB is given, and the area APBMA is a maximum.

Let QM = $y$, then PM = $f(y)$, where $f$ is a known function.

Also length AQB = $\backslash int\; \backslash sqrt\{1+p^2\}\; dx$, and area APB = $\backslash int\; f(y)\; dy$, the limits of both integrals being the same.

and by formula (4) $f(y)\; +\; a\; \backslash sqrt\{1+p^2\}\; =\; \backslash frac\{ap^2\}\{\backslash sqrt\{1+p^2\}\}\; +\; c$,

the differential equation to the curve AQB.

If $y\; =\; \backslash phi(x)$ be the solution of this equation, $y\; =\; f\backslash \{\backslash phi(x)\backslash \}$ is the equation to the curve APB.

Cor. If $f(y)\; =\; y$, we get $a^2\; =\; (c-y)^2\; +\; (c\text{'}-y)^2$, the equation to a circle.

Prob. 6. To find the curve which, with a given length, contains between its chord and its arc the greatest possible area.

By cor. to last problem the equation is $a^2\; =\; (c-y)^2\; +\; (c\text{'}-y)^2$.

Let the origin be at A and AB = $2r$; then since $y=0$, both when $x=0$ and $x=2r$, we get

And from the equation to the limits $(\backslash gamma\_1\; -\; P\_1\; p\_1)\; dx\_1\; =\; 0$, or $\backslash gamma\_1\; -\; P\_1\; p\_1\; =\; 0$ $\backslash therefore\; c=0$; and the curve required is the semicircle.

Prob. 7. Given the length of the curve contained between two points in a horizontal line; required its nature, that the centre of gravity of the arc may be the lowest possible. Here we have to make $\backslash int\; x\; \backslash sqrt\{1+p^2\}\; dx$ a maximum, whilst $\backslash int\; \backslash sqrt\{1+p^2\}\; dx$ is constant.

the integral of which is $y\; =\; c\; \backslash log.\; (x+a\; +\; \backslash sqrt\{(x+a)^2\; -\; c^2\})\; +\; c$, the equation to the catenary.

The solution of more complicated problems is not consistent with our limits. For the investigation of formulae in cases where $\backslash gamma$ involves an integral, or where it is given by a differential equation, the reader is referred to Woodhouse's Treatise. Considerations on the mode of distinguishing a maximum from a minimum will be found in Lagrange's Theorie des Fonctions Analytiques.

VARIATION of the Compass. See MAGNETISM.