Page:EB1911 - Volume 27.djvu/941

 the multiplier to problems in which the variations are which distinguishes the composers who seem to know their theme from those who do not.

VARIATIONS, CALCULUS OF, in mathematics. The calculus of variations arose from the attempts that were made by mathematicians in the 17th century to solve problems of which the following are typical examples. (i) It is required to determine the form of a chain of given length, hanging from two fixed points, by the condition that its centre of gravity must be as low as possible. This problem of the catenary was attempted without success by Galileo Galilei (1638). (ii) The resistance of a medium to the motion of a body being assumed to be a normal pressure, proportional to the square of the cosine of the angle between the normal to the surface and the direction of motion, it is required to determine the meridian curve of a surface of revolution, about an axis in the direction of motion, so that the resistance shall be the least possible. This problem of the solid of least resistance was solved by Sir Isaac Newton (1687). (iii) It is required to find a curve joining two fixed points, so that the time of descent along this curve from the higher point to the lower may be less than the time along any other curve. This problem of the brachistochrone was proposed by John (Johann) Bernoulli 1696).

The contributions of the Greek geometry to the subject consist of a few theorems discovered by one Zenodorus, of whom little is known. Extracts from his writings have been preserved in the writings of Pappus of Alexandria and Theon of Smyrna. He proved that of all curves of given perimeter the circle is that which encloses the largest area. The problems from which the subject grew up have in common the character of being concerned with the maxima and minima of quantities which can be expressed by integrals of the form $\int_{x_0}^{x_1}\text{F}(x, y, y')dx,$

in which y is an unknown function of x, and F is an assigned function of three variables; viz. x, y, and the differential coefficient of y with respect to x, here denoted by y′; in special cases x or y may not be explicitly present in F, but y′ must be. In any such problem it is required to determine y as a function of x, so that the integral may be a maximum or a minimum, either absolutely or subject to the condition that another integral or like form may have a. prescribed value. For example, in the problem of the catenary, the integral $\int_{x_0}^{x_1}y(1+ y'^2)\tfrac{1}{2}dx,$

must be a minimum, while the integral

$\int_{x_0}^{x_1}(1+ y'^2)\tfrac{1}{2}dx,$

has a given value. When, as in this example, the length of the sought curve is given, the problem is described as isoperimetric. At the end of the first memoir by James (Jakob) Bernoulli on the infinitesimal calculus (1690), the problem of determining the form of a flexible chain was proposed. Gottfried Wilhelm Leibnitz gave the solution in 1691, and stated that the centre of gravity is lower for this curve than for any other of the same length joining the same two points. The first step towards a theory of such problems was taken by James Bernoulli (1697) in his solution of the problem of the brachistochrone. He pointed out that if a curve, as a whole, possesses the maximal or minimal property, every part of the curve must itself possess the same property. Beyond the discussion of special problems, nothing was attempted or many years.

The first general theory of such problems was sketched by Leonhard Euler in 1736, and was more fully developed by him in his treatise Methodus inveniendi. . . published in 1744. He generalized the problems proposed by his predecessors by admitting under the sign of integration differential coefficients of order higher than the first. To express the condition that an integral of the form

$\int_{x_0}^{x_1}\text{F}(x, y, y', y'', . . . y^{n)})dx$

may be a maximum or minimum, he required that, when y is changed into y+u, where u is a function of x, but is everywhere “infinitely” small, the integral should be unchanged. Resolving the integral into a sum of elements, he transformed this. condition into an equation of the form

ux [$∂F⁄∂y$ −. . . ] ＝0,

and he concluded that the differential equation obtained by equating to zero the expression in the square brackets must be satisfied. This equation is in general of the 2nth order, and the 2n arbitrary constants which are contained in the complete /primitive-must be adjusted to satisfy the conditions that y, y′, y″,. . . y(n−1) have given values at the limits of integration. If the function y is required also to satisfy the condition that another integral of the same form as the above, but containing a function instead of F, may have a prescribed value, Euler achieved his purpose by replacing F in the differential equation by F+, and adjusting the constant so that the condition may be satisfied. This artifice is known as the isoperimetric rule or rule of the undetermined multiplier. Euler illustrated his methods by a large number of examples.

The new theory was provided with a special symbolism by Joseph Louis der la Grange (commonly called Lagrange) in a series of memoirs, published in 1760–62. This symbolism was afterwards adopted by Euler (1764), and Lagrange is generally regarded as the founder of the calculus of variations. Euler had been under the necessity of resolving an integral into a sum of elements, recording the magnitude of the change produced in each element by a slight change in the unknown function, and thence forming an expression for the total change in the sum under consideration. Lagrange proposed to free the theory from this necessity. Euler had allowed such changes in the position of the curve, along which the integral, to be made a maximum or minimum, is taken, as can be produced by displacement parallel to the axis of ordinates. Lagrange admitted a more general change of position, which was called variation. The points of the curve being specified by their co-ordinates, x, y, z, and differentiation along the curve being denoted, as usual, by the symbol d, Lagrange considered the change produced in any quantity Z, which is expressed in terms of x, y, z, dx, dy, dz, d2x,  when the co-ordinates x, y, z are changed by “infinitely” small increments. This change he denoted by BZ, and regarded as the variation of Z. He expressed the rules of operation with by the equations dZ＝dZ, ∫Z＝∫Z.

By means of these equations ∫Z can be transformed by the process of integration by parts into such a form that differentials of variations occur at the limits of integration only, and the transformed integral contains no differentials of variations. The terms at the limits and the integrand of the transformed integral must vanish separately, if the variation of the original integral vanishes. The process of freeing the original integral from the differentials of variations results in a differential equation, or a system of differential equations, for the determination of the form of the required curve, and in special terminal conditions, which serve to determine the constants that enter into the solution of the differential equations. Lagrange's method lent itself readily to applications. of the generalized principle of virtual velocities to problems, of mechanics, and he used it in this way in the Mécanique analytique (1788). The terminology and notation of mechanics are still largely dominated. by these ideas of Lagrange, for his methods were powerful and effective, but they are rendered obscure by the use of “infinitely” small quantities, of which, in other departments of mathematics, he subsequently became an uncompromising opponent. The same

ideas were: applied by Lagrange himself, by Euler, and by other mathematicians to various extensions of the calculus of variations. These include problems concerning integrals of which the limits are variable in accordance with assigned conditions, the extension of Euler’s rule of restricted by conditions of various types, the maxima and minima of integrals involving any number of dependent variables, such as are met with in the formulation of the dynamical Principle of Least Action, the maxima and minima of double and multiple integrals. In all these cases Lagrange's methods have been applied successfully to obtain the differential equation, or system of differential equations, which must be satisfied if the integral in question is a maximum or a minimum. This equation, or equations, will be referred to as the principal equation, or principal equations, of the problem.

The problems and method of the calculus admit of more exact formulation as follows: We confine our attention to the case where the sought curve is plane, and the function F contains no differential coefficients of order higher than the first. Then the problem is to determine a curve joining two fixed points (x0, y0) and (x1, y1) so that the line integral

$\int_{x_0}^{x_1}\text{F}(x, y, y')dx$

taken along the curve may be a maximum or a minimum. When it is said that the integral is a minimum for some curve, it is meant that it must be possible to mark a finite area in the plane of (x, y), so that the curve in question lies entirely within this area, and the integral taken along this curve is less than the integral taken along any other curve, which joins the same two points and lies entirely within the delimited area. There is a similar definition for a maximum. The word extrernurn is often used to connote both maximum and minimum. The problem thus posed is known as the First Problem of the Calculus of Variations. If we begin with any curve