Calculus of Variations/CHAPTER XIII

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  • 179 The general problem stated.
  • 180 Existence of substitutions by which one integral remains unchanged while the other is caused to vary. An exceptional case.
  • 181 Case of two variables. Convergence of the series that appear.
  • 182 The nature of the substitutions that have been introduced.
  • 183 Formation of certain quotients which depend only upon the nature of the curve.
  • 184 Generalization, in which several integrals are to retain fixed values.
  • 185 The quotient of two definite integrals being denoted by , it is shown that has the same constant value for the whole curve.
  • 186 The differential equation .
  • 187 Extension of the theorem of Article 97.
  • 188 Discontinuities, etc.
  • 189 The second variation: the three conditions formulated in Article 135 are also necessary here.

Article 179.
The nature of many problems whicli arise in the Calculus of Variations presents subsidiary conditions which limit the arbitrariness that we have hitherto employed in the indefinitely small variations of the analytical structure. Such problems are the most difficult and at the same time the most interesting that occur. These last conditions which enter into the requirement for a maximum or a minimum are in general of a double nature. On the one hand, it may be proposed that among the variables there are to exist equations of condition, as indicated in Arts. 176 and 177. On the other hand, we may require that the maximum or the minimum in question satisfy a further condition, viz., it must cause another given integral to have a prescribed value. Such cases are usually called Relative Maxima and Minima.

If we limit our discussion to the region of two variables, then the problem which we have to consider may be expressed as follows (cf. Art. 17):

Let and be two functions of the same nature as the function hitherto treated. The variables and are to be so determined as one-valued functions of that the curve defined through the equations will cause the integral

to be a maximum or a minimum, while at the same time for the same equations the integral

will have a prescribed value; that is, for every indefinitely small variation of the curve for which the second integral retains its sign unaltered, the first integral, according as a maximum or a minimum is to enter, must be continuously smaller or continuously greater than it is for the curve .

Article 180.
We must first show that it is possible to represent analytically the variations of a curve for which the integral retains a constant value.

In the place of the variables let us make the substitution . The variation of the second integral is accordingly

where denotes that the terms within the brackets are of the second and higher dimensions in .

We have so to determine and ; that . For this purpose we write

where are arbitrary constants and the functions are functions similar to the quantities of the preceding Chapters and vanish for and . Now write


Hence, from 3) we have

If we write

it follows that

The functions are completely determined as soon as definite values are given to ; and, in order that , it is necessary that

If any of the quantities , for example , are different from zero, we are able to express in a power-series of the remaining 's, when these quantities have been chosen sufficiently small.[1] The equation may consequently be satisfied for sufficiently small systems of values of the 's.

Substitute one of these systems of values in 4) and it is seen that indefinitely small variations of the curve exist for which the integral remains unaltered. These variations may be analytically represented (see the next Article).

This proof is deficient in the case where all the quantities are zero for all values of , however may have been chosen. When this is the case, must be zero along the whole curve. But this is one of the necessary conditions that the integral have a maximum or a minimum value.

If, then, for the curve which is derived through the solution of the differential equation there also enters a maximum or a minimum value of the integral and consequently , it is in general not possible so to vary the curve that the second integral remains unaltered.

This case is excluded from the present discussion, and is left for special investigation in each particular problem.

Article 181.
Let us limit ourselves for the present to the simplest case where

and if we denote the integrals in the expansion of that are associated with the coefficients by , the equation correresponding to (A) of the last article is

which series we suppose convergent for sufficiently small values of and .

Suppose next we express in terms of by the series

Then, when this value of is substituted in , by equating the coefficients of the different powers of to zero, we have


Hence, denoting the quotients by , where , we have


Further, the equation may be written

Let us compare this series with the series

Suppose from this series we have expressed in terms of in the form

where the s have been derived from the coefficients of powers of and as the s in are formed from the coefficients in .

The series is convergent for

If, then, the coefficients of are in absolute value less than the corresponding coefficients in , the coefficients in are less in absolute value than the coefficients in , and therefore the series is convergent.

Now the coefficients of in and are respectively


where the symbol denotes . Hence for sufficiently small values of and , if


the series is convergent, and when substituted in the expression for causes this expression to vanish.

Article 182.
The expression for as a function of is had from the relation

Hence, it follows that


Of the two roots we choose the one with the lower sign in order that equal zero with . This root may be written

It is seen that the expression under the radical is finite, continuous and one-valued for values of such that


Article 183.
Returning to the substitutions

we assume that the functions become zero at the endpoints (or limits) of the curve and are so chosen that does not vanish within the limits of integration. We have then at once from the power-series

where the power-series vanishes with .

From this we have

If we subject the integral to the same variation, we have [cf. formula ]

and consequently

If then, the integral is to have a maximum or a minimum value, it is necessary that

be equal to zero.

We have, therefore, the necessary condition

From this it is seen that the quotient , is independent of the arbitrary functions , since it does not vary if we write for as functions of other functions . Consequently it follows that the value of the above quotient depends only upon the nature of the curve .

Article 184.
We might generalize the problem treated above by requiring the curve which minimizes or maximizes the integral

while at the same time the following integrals have a prescribed value:


the functions being of the same nature as the function defined in Chapter I.

We must now consider the deformation of the curve caused by the variations

We have, then, if we write , and suppose that the 's and 's vanish for and


By means of the last equations, if the determinant

is different from zero, we may, for sufficiently small values of , express these quantities as convergent power-series in [2]

These power-series when substituted in cause it to have the form


In order that the integral have a maximum or a minimum value, it is therefore necessary that

This determinant, when expanded, may be written in the form

where is the first minor of in the determinant .

Hence, as before (cf. Art. 79, where we had ), we have here

Article 185.
Similarly, if in Art. 183 we denote the quotient by and then give to and their values, we have

From this it follows that

We may prove a very important theorem regarding the constant , viz: -it has one and the same value for the whole curve; i. e., we always have the same value of , whatever part of the curve we may vary. Consider the values of laid off on a straight line, and suppose that the constant has a definite value for, say, the interval which also corresponds to a certain portion of curve. This value (see Art. 183) is independent of the manner in which the portion of curve has been varied. Next consider an interval which includes the interval ; then, there belongs to all the possible variations of the interval , also that variation by which and remain unchanged and only , varies. As has a definite value for this interval and is independent of the manner in which the curve has been varied, it must have the same value for .

Article 186.
The differential equation is the same as the one we would have if we require that the integral

have a maximum or a minimum value, where is written for the function

Through this differential equation (See Art. 90) ; and are expressible in terms of and and two constants of integration and in the form

The curve represented by these equations is a solution of the problem, when indeed a solution is possible.

Article 187.
We prove next a very important theorem which often gives a criterion whether a sudden change in direction can take place or not within a stretch where the variation is unrestricted (cf. Art. 97). Suppose that on a position , where the variation is unrestricted, a sudden change in direction is experienced. On either side of take two points and so near to that within the intervals and a similar discontinuity in change of direction is not had. Among the possible variations there is one such that the whole curve remains unchanged except the interval , which is, of course, varied in such a way that the integral retains its value. The variation of the integral depends then only upon the variation of the sum of integrals

We cause a variation in the stretch by writing

where we assume that


are all zero for and
are zero for

We may then always determine as a power-series in so that .

If by we denote an expression of the form , we have (Art. 79)

If the curve minimizes or maximizes the integral , it is necessary that the coefficient of on the right-hand side of the above expression be zero. Since for unrestricted variation, it follows from the assumption (A) that

If in the assumptions (A) we assume for that and , we have an analogous equation for .

It therefore follows (cf. Art. 97) that

We have then the theorem : Along those positions which are free to vary of the curve which satisfies the differential equation , the quantities and vary everywhere in a continuous manner, even on such positions of the curve where a sudden change in its direction takes place.

Article 188.
It is obvious that these discontinuities may all be avoided, if we assume that vanish at such points. This we may suppose has been done. We may also impose many other restrictions upon the curve ; for example, that it is to go through certain fixed points, or that it is to contain certain given portions of curve, or that it is to pass through a certain limited region. In all these cases there are points on the curve which cannot vary in a free manner. But whatever condition may be imposed upon the curve, the following theorem is true.

All points which are free to vary and there always exist such points must satisfy the differential equation , and for all such points the constant has the same value.

Article 189.
The second variation. We assume that the variations at the limits and at all points of the curve where there is a discontinuity in the direction, vanish. We also suppose that the variations have been so chosen that .

We then have (cf. Art. 115):

and consequently


it follows that

This last integral may be written at once (Art. 119) in the form

where is determined from the differential equation (Art. 118)

It follows here as a necessary condition for the existence of a maximum or a minimum that for all portions of the curve at which there is free variation, must in the first case be everywhere negative' and in the second case everywhere positive' and must also be different from 0 and . In order that this transformation of the integral be possible the equation must admit of being integrated in such a way that is different from zero on all portions of curve, which vary freely (Art. 128).

We shall determine in Chapter XVII whether the three necessary conditions thus formulated are also sufficient for a maximum or a minimum value of the integral . By means of the example in the next Chapter, we shall also show that if there exists a curve, for which the first integral has a maximum or a minimum value while the second integral retains a given value, then the curve is determined through the three conditions, which are the same here as those formulated in Art. 135. The behavior of the -function is then decisive regarding whether there in reality exists a maximum or a minimum.

  1. Cf. Lectures on the Theory of Maxima and Minima, etc., p. 20.
  2. Cf. Lectures on the Theory of Maxima and Minima of Functions of Several Variables, p. 21.