VARIATIONS, CALCULUS OF. 1. The object of this calculus is the discovery of the form of a function which shall fulfill 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,

\[ \frac{PQ}{\text{velocity at } Q} + \frac{QR}{\text{at } R} = \frac{PS}{\text{at } Q} + \frac{SR}{\text{at } R} \]

ultimately;

or

\[ \frac{PQ - PS}{\sqrt{NS}} = \frac{SR - QR}{\sqrt{OR}}, \quad \text{or} \quad \frac{Qe}{\sqrt{NS}} = \frac{Sc}{\sqrt{OR}} \]

ultimately;

that is,

\[ \frac{\cos \angle SQe}{\sqrt{NS}} = \frac{\cos \angle QSe}{\sqrt{OR}} \]

ultimately;

or

\[ \frac{\sin \angle}{\text{do.}} = \frac{\text{curve makes with vertical at } Q}{\text{do.}} = \frac{\sqrt{NS}}{\sqrt{OR}}, \]

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 1741, when Euler published his treatise, entitled Methodus invendi Lineas Curvas proprietas maximae minime 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 δ 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 Théorie 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 dx and dy.

Let \( u = f(y)dx \) be the formula under consideration; in which γ involves x, y, \( \frac{dy}{dx}, \frac{d^2y}{dx^2}, \ldots \).

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

Then \( \delta u = (f(y) + \delta y)(dx + \delta x) - f(y)dx, \)

\[ = \delta f(y)dx + f(y)\delta x, \]

\[ = \delta f(y)dx + \gamma \delta x - f(x)\delta y. \]

Now \( \delta f(y)dx = \delta x(Mdx + Ndy + Pdp + \ldots), \)

and \( \delta f(y)dx = \delta x(Mdx + Ndy + Pdp + \ldots), \)

\[ \therefore \delta u = \gamma \delta x + \int \left[ N(d\delta y - \delta dy) + P(d\delta p - \delta dp) + \ldots \right] dx. \]

\[ = \gamma \delta x + \int \left[ N(\delta y - p\delta x) + P(2p - q\delta x) + \ldots \right] dx. \]

Also if \( \delta y - p\delta x \) be denoted by ω, we have

\[ \delta p - q\delta x = \frac{d(y + \delta y)}{dx + \delta x} - \frac{dy}{dx} - \frac{dp}{dx} \delta x, \]

\[ = \frac{d\delta y}{dx} - p\frac{d\delta x}{dx} - \frac{dp}{dx} \delta x, \]

\[ = \frac{d\delta y}{dx} - \frac{d\delta x}{dx}(p\delta x) = \frac{d}{dx}(\delta y - p\delta x) = \frac{d\omega}{dx}; \]

hence putting p for y and q for p, we obtain

\[ \delta q - r\delta x = \frac{d}{dx}(\delta p - q\delta x) = \frac{d}{dx} \frac{da}{dx} = \frac{d^2a}{dx^2}, \]

&c. = &c.

By substitution,

\[ \delta u = \gamma \delta x + \int \left[ N\omega + P\frac{da}{dx} + Q\frac{d^2a}{dx^2} + \ldots \right] dx. \]

But \( \int P\frac{da}{dx} dx = P\omega - \int \omega \frac{dP}{dx} dx, \)

\[ \int Q\frac{d^2a}{dx^2} dx = Q\frac{da}{dx} - \omega \frac{dQ}{dx} + \int \omega \frac{d^2Q}{dx^2} dx, \]

&c. = &c.

\[ \therefore \delta u = \gamma \delta x + \omega(P - \frac{dQ}{dx} + \frac{d^2R}{dx^2} - &c.) \]

\[ + \frac{da}{dx}(Q - \frac{dR}{dx} + \ldots) \]

\[ + &c. &c. \]

\[ + \int \omega(N - \frac{dP}{dx} + \frac{d^2Q}{dx^2} - &c.) dx. \]

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

Let \( x_1, y_1, x_2, y_2 \) be the values of x and y at the two limits; then the whole value of \( \delta u \) between the limits is,

\[ \delta u = \gamma \delta x_1 - \gamma \delta x_2 + \omega(P_1 - &c.) - \omega(P_2 - &c.) \]

\[ + &c. + \int \omega(N - \frac{dP}{dx} + \frac{d^2Q}{dx^2} - &c.) dx. \]

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

But this quantity consists of two parts; the one an integrated expression, depending only on the values of \( \delta x, \delta y, \ldots \); the other an unintegrated expression, depending on the general values of \( \delta x, \delta y \). 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 ω, without making the two parts separately zero.

Our conditions are therefore,

\[ N - \frac{dP}{dx} + \frac{d^2Q}{dx^2} - &c. = 0. \quad (1) \]

\[ \gamma \delta x_1 - \gamma \delta x_2 + \omega(P_1 - &c.) - \omega(P_2 - &c.) = 0. \quad (2) \]

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 \( \delta x \) and \( \delta y \). This will explain why the unintegrated part of the expression has been reduced to the form in which we left it. Had ω been implicated with x and y, But \( P \), in this case, is

\[ \frac{p}{\sqrt{1+p^2}} \cdot \frac{\sqrt{x}}{x} ; \quad \frac{ds}{dy} = \frac{1}{c \sqrt{x}}, \]

the same result as before, with the exception that \( x \) and \( y \) are interchanged. The solution of this equation shows 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.

The resistance varies as

\[ \int \frac{3y^2}{1+p^2} dx; \quad \text{by formula (4)} \]

\[ \frac{3yp^2}{1+p^2} = \frac{2yp^4}{(1+p^2)^2} + c, \]

or \( 0 = \frac{2yp^3}{(1+p^2)^2} + c. \)

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 \( y \) is here \( \sqrt{1+p^2} \) where, as the line on which the distance is measured is a right line, \( p = c. \)

But by formula (2)

\[ \gamma_1 \delta x_1 - \gamma_2 \delta x_2 + P_{1w} - P_{2w} = 0, \quad \text{when } \delta x_1, \delta x_2 \text{ are indefinitely small.} \]

Also, if the curves have no connexion, we may make \( \delta x_1, \delta y_1 \) assume what values soever we please, without affecting \( \delta x_2, \delta y_2 \); we must therefore have separately

\[ \gamma_1 \delta x_1 + P_{1w} = 0, \quad \text{and } \gamma_2 \delta x_2 + P_{2w} = 0. \]

The latter gives \( \sqrt{1+p^2} \delta x_2 + \frac{P_{2w}}{\sqrt{1+p^2}} (2y_2 - p \delta x_2) = 0, \)

or \( \delta x_2 + P_{2w} = 0, \quad \text{or } \frac{\delta y_2}{\delta x_2} = -\frac{1}{P_{2w}} = -\frac{1}{c}, \quad \text{when } \delta x_2 \text{ and } \delta y_2 \text{ are indefinitely diminished.}

But, since the measurement must begin in the first curve, the limit of \( \frac{\delta y_2}{\delta x_2} \) expresses the tangent of the angle in which the tangent to the first curve cuts the axis. Our equation, then, shows 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:

then \( t = \frac{1}{\sqrt{2g}} \int \frac{\sqrt{1+p^2}}{\sqrt{y-h}} dx, \) from which (by Prob. 1.)

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):

\[ \gamma_1 \delta x_1 - \gamma_2 \delta x_2 + P_{1w} - P_{2w} + \delta h \int \frac{dy}{dx} dx = 0. \]

The problem admits of several cases.

1. If the body starts from a given point in the first curve, that curve performs no part in the problem. Here \( \delta x_1, \delta x_2 \) and \( \delta h \) are all zero.

\[ \gamma_1 \delta x_1 + P_{1w} (\delta y_1 - p \delta x_1) = 0. \]

But \( \gamma_1 - P_{1w} = c, \quad \text{(see Prob. 1.)} \)

\[ \frac{1}{\sqrt{1+p^2}} \cdot \frac{\sqrt{y-h}}{y-h} = c, \quad \text{and } P_{1w} = \frac{P_{1w}}{\sqrt{1+p^2}} \cdot \frac{\sqrt{y-h}}{y-h} = P_{1w}. \]

\[ \delta x_1 + P_{1w} \delta y_1 = 0; \quad \text{or } \frac{dy}{dx} \text{ for the second curve, at the} \] point in which the cycloid cuts it, is equal to \( \frac{1}{p} \); which shows 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

\[ \gamma_1 \delta x_1 - \gamma_2 \delta x_2 + P_1 \omega_1 - P_2 \omega_2 = 0, \]

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, \( h = y_a \), and

\[ \frac{dy}{dh} = -N = -\frac{dP}{dx} (\because N = \frac{dP}{dx} = 0); \]

hence our equation becomes

\[ \gamma_1 \delta x_1 - \gamma_2 \delta x_2 + P_1 \omega_1 - P_2 \omega_2 = 0, \]

or

\[ \gamma_1 \delta x_1 + P_1 \omega_1 = 0, \quad \gamma_2 \delta x_2 + P_2 \omega_2 = 0. \]

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

\[ (\gamma_1 - P_1 p_1) \delta x_1 + P_1 \delta y_1 = 0, \quad \text{or} \quad c + P_1 \frac{\delta y_1}{\delta x_1} = 0. \]

Also the first equation is \( c + P_1 \frac{\delta y_1}{\delta x_1} = 0 \).

\[ \therefore \frac{\delta y_1}{\delta x_1} = \frac{\delta y_2}{\delta x_2}; \]

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 minimum absolutely, but only one consistent with the further condition that another integrated function shall not change its value. Although, therefore, \( \int f(x) dx \) is actually zero, yet \( \delta x \) and \( \delta y \) are not, as in our previous investigations, any quantities whatever, but only such as will consist with the required condition that \( \int \sqrt{1 + p^2} dx \) shall also be zero. To Euler we owe the idea of substituting for the quantity \( u \), which is made to vary, not \( f(x) \), as in other cases, but \( f(x) + af(x) \); \( f(x) \) being the quantity which is to remain unchanged. By this substitution we determine the relation between \( x \) and \( y \), which renders \( y + av \) 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 \( f(x) \) 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 \( AQ \) is given, and the area \( APBMA \) is a maximum.

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

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

\[ \therefore y = f(y) + a \sqrt{1 + p^2}, \]

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

or \( \{c - f(y)\} \sqrt{1 + p^2} = a, \)

the differential equation to the curve \( AQB \).

If \( y = \phi(x) \) be the solution of this equation, \( y = \int [\phi(x)] \) is the equation to the curve \( APB \).

Cor. If \( f(y) = y \), we get \( a^2 = (c - y)^2 + (c' - x)^2 \), 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 \( a^2 = (c - y)^2 + (c' - x)^2 \).

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

\[ y^2 - 2cy = 2rx - x^2. \]

And from the equation to the limits \( (\gamma_1 - P_1 p_1) \delta x = 0 \), or \( \gamma_1 - P_1 p_1 = 0 \), \( c = 0 \); 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 \( \int \sqrt{1 + p^2} dx \) a maximum, whilst \( \int \sqrt{1 + p^2} dx \) is constant.

\[ \therefore y = x \sqrt{1 + p^2} + a \sqrt{1 + p^2}, \quad \text{and} \quad \frac{dP}{dx} = 0, \quad P = c, \]

or \( \frac{p(x+a)}{\sqrt{1 + p^2}} = c, \)

the integral of which is \( y = c \log (x + a + \sqrt{(x+a)^2 - c^2}) + e \), 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 \( y \) 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 Théorie des Fonctions Analytiques.

Variation of the Compass. See Magnetism.