CHAPTER VIII: THE SECOND VARIATION; ITS SIGN DETERMINED BY THAT OF THE FUNCTION .
- 111 Nature and existence of the substitutions introduced.
- 112 The total variation.
- 113,114 The second variation of the function .
- 115 The second variation of the integral . The sign of the second variation in the determination of maximum or minimum values.
- 116 Discontinuities.
- 117 The sign of the second variation made to depend upon that of .
- 118 The admissibility of a transformation that has been made. The differential equation .
- 119 A simple form of the second variation.
- 120 A general property of a linear differential equation of the second order.
- 121 The second variation and the function . The function cannot change sign and must be different from and in order that there may be a maximum or a minimum.
The substitution , for , causes any point of the original curve to move along a straight line, which makes an angle with the -axis whose tangent is .
This deformation of the curve is insufiEcient, if we require that the point move along a curve other than a straight line.
To avoid this inadequacy we make the more general substitution (by which the regular curve remains regular):
where, like , in our previous development (Art. 75), the quantities , , , are functions of , finite, continuous one-valued and capable of being differentiated (as far as necessary) between the limits . These series are supposed to be convergent for values of such that .
That such substitutions exist may be seen as follows:
Since the curve is regular, the coordinates of consecutive points to and may be expressed by series in the form, say,
where the coefficients of the powers of are constants and the series are convergent.
Suppose, now, that we seek to determine the functions of
such that for and , the expressions (C) will be the same as (A) and (B).
This may be done, for example, by writing
and then determine , , , in such a way that
- ; ,
- ; .
From this it is seen that
- , etc.
In the same way we may determine quadratic expressions in for , , etc.
The substitutions thus obtained are of the nature of those which we have assumed to exist, and may evidently be constructed in an infinite number of different ways.
Making the above substitutions in the integral
it is seen that
By Taylor's Theorem we have
The coefficient of in this expression is the integrand of and is zero; while the coefficient of involves terms that are the first partial derivatives of , and also those that are the second partial derivatives of .
The first partial derivatives of that belong to this coefficient, when put under the integral sign, may be written in the form
(see Art. 79), and this expression is also zero, if we suppose that the end-points remain fixed.
The coefficient of in the preceding development of by Taylor's Theorem is, neglecting the factor , denoted by .
We have then
The subscripts may now be omitted and the formula simplified by the introduction of the function , which (Art. 73) was defined by the relations:
- , , ;
and by introducing the new notation:
- , (owing to the equation ), ;
where , are used for ,.
We have then
To get an exact differential as a part of the right-hand member of this formula, we write
an expression which, differentiated with respect to , becomes
We further write
where (see Art. 81) is, neglecting the factor , the amount of the sliding of a point of the curve in the direction of the normal.
Differentiating with respect to , we have
from which it follows that
Then the expression for the second variation becomes
If further we write in this expression
- , , ,
we have finally
It follows from 3) that
Owing to the homogeneity of the function (Chap. IV), it is seen from Euler's Theorem that
In a similar manner we have
Differentiating with regard to , the above expression becomes
which, owing to 3) is
or from 6)
In an analagous manner it may be shown that
From these expressions we have at once
where is the factor of proportionality.
It follows that
The quantity is defined through these three equations and plays an essential role in the treatment of the second variation.
Owing to the relation 7)
- becomes ,
The second variation of the integral has therefore the form
We suppose that the end-points are fixed so that at these points , and we further assume that the curve subjected to variation consists of a single regular trace, along which then
is everywhere continuous, so that
Consequently the above integral may be written
If the integral is to be a maximum or a minimum for the curve , it is necessary, when the curve is subjected to an indefinitely small variation, that the variation , which is caused to exist therefrom, have always the same sign, in whatever manner , are chosen; and consequently the second variation must have continuously the same sign as .
We have repeatedly seen that
and for any other value of for example, ,
If, further, is negative while is positive, then we may take so small that the sign of depends only upon the first term on the right in the above expansion, and consequently is negative. Therefore the integral cannot be a maximum or a minimum, since the variation of it is first positive and then negative.
Hence, neglecting for a moment the case when , we have the following theorem:
If the integral is to be a maximum or a minimum, its second variation must be continuously negative or continuously positive.
When vanishes for all possible values of , , it is necessary also that vanish, since the integral is to be a maximum or a minimum, and, as in the Theory of Maxima and Minima, we would then have to investigate the fourth variation. In this case the conditions that have to be satisfied are so numerous that a mathematical treatment is very complicated and difficult.
Hence, it is seen that after the condition is satisfied, it follows that
- for the possibility of a maximum, must be negative, and
- for the possibility of a minimum, must be positive.
These conditions are necessary, but not sufficient.
In Art. 75 we assumed that ,,, were continuous functions of between the limits . Owing to the assumed existence of ,, we must presuppose the existence of the second derivatives of and with respect to (see Art. 23). From this it also follows that the radius of curvature must vary in a continuous manner. These assumptions have been tacitly made in the derivation of the equation 8) in the preceding article. We shall now free ourselves from the restriction that and are continuous functions of , retaining, however, the assumptions regarding the continuity of the quantities .
The theorem that and vary in a continuous manner for ox oy the whole curve (Art. 97) in most cases gives a handy means of determining the admissibility of assumptions regarding the continuity of and . If, at certain points of the curve , and are not continuous, it is always possible to divide the curve into such portions that and are continuous throughout each portion. Yet we cannot even then say that and are continuous within such a portion, as has been assumed to be true in the above development. If, however, and within such a portion of curve are discontinuous, we have only to divide the curve into other portions so that within these new portions and no longer suffer any sudden springs. In each of these portions of curve the same conclusions may be made as before in the case of the whole curve, and consequently the assumption regarding the continuous change of , throughout the whole curve is not necessary. But if we had limited ourselves to the consideration of a part of the curve in which vary in a continuous manner, the continuity of , in the integration of the integral
would have been assumed. These assumptions need not necessarily be fulfilled, since the variation of the curve is an arbitrary one, and it is quite possible that such variations may be introduced, where , become discontinuous, as often as we please. We may, however, drop these assumptions without changing the final results, if only the first named conditions are satisfied. Since the quantities , , depend only upon , and since these quantities are continuous, it follows that the introduction of the integral the form given above is always admissible. For if , were not continuous for the whole trace of the curve, which has been subjected to variation, we could suppose that this curve has been divided into parts, within which the above derivatives varied in a continuous manner, and the integral would then become a sum of integrals of the form
where are the coordinates of the points of division of corresponding values of . But since , vary in a continuous manner, we have through the summation of these quantities exactly the same expression
as before. The quantities , are also found under the sign of integration in the right-hand side of 8); but owing to the conception of a definite integral, we may still write it in this form even when these quantities vary in a discontinuous manner ; however, in performing the integration, we must divide the integral corresponding to the positions at which the discontinuities enter into partial integrals. Therefore, we see that the possible discontinuity of , remains without influence upon the result, if only are continuous. Consequently any assumptions regarding the continuity of , are superfluous; however, in an arbitrarily small portion of the curve which is subjected to variation, the quantities and must not become discontinuous an infinite number of times since such variation of the curve has been necessarily, once for all excluded.
Following the older mathematicians, Legendre, Jacobi, etc., we may give the second variation a form in which all terms appearing under the sign of integration will have the same sign (plus or minus).
To accomplish this, we add an exact differential at under the integral sign in 8), and subtract it from , the integral thus becoming
The expression under the sign of integration is an integral homogeneous quadratic form in and . We choose the quantity so that this expression becomes a perfect square; that is,
We shall see that it is possible to determine a function , which is finite one-valued and continuous within the interval , and which satisfies the equation 9). The integral 10) becomes accordingly, if the end-points remain fixed,
Hence the second variation has the same sign as , and it is clear that for the existence of a maximum must be negative, and for a minimum this function must be positive within the interval and in case there is a maximum or a minimum, cannot change sign within this interval.
This condition is due to Jacobi. Legendre had previously concluded that we have a maximum when a certain expression corresponding to was negative, and a minimum when it was positive. It is questionable whether the diflferential equation for is always integrable. Following Jacobi we shall show that such is the case.
Before we go farther, we have yet to prove that the transformation, which we have introduced, is allowable. In spite of the simplicity of the equation 9) we cannot make conclusions regarding the continuity of the function v, which is necessary for the above transformation. ¢ It is therefore essential to show that the equation 9) may be reduced to a system of two linear differential equations, which may be reverted into a linear differential equation of the second order, since for this equation we have definite criteria of determining whether a function which satisfies it remains finite and continuous or not.
where and are continuous functions of , and within the interval .
Equation 9) becomes then
Since one of the functions , may be arbitrarily chosen, we take so that
then, since , we have
From 11) and 12) it follows that
where and are to be considered as given functions of . We shall denote this difFerential equation by . After has been determined from this Equation, may be determined from 11), and from we have as a definite function of .
The expression which has been derived for seems to contain two arbitrary constants, while the equation 9) has only one. The two constants in the first case, however, may be replaced by one, since the general solution of 13) is
and hence from 11)
an expression which depends only upon the ratio of the two constants.
It follows from the above transformation that
but this transformation has a meaning only when it is possible to find a function within the interval which is different from zero, and which satisfies the differential equation .
If we have a linear differential equation of the second order
and if and are a fundamental system of integrals of this equation, then we have the well known relation due to Abel (see Forsyth's Differential Equations, p. 99)
If , then we would have , and the system is no longer a fundamental system of integrals. This determinant can become zero only at such positions for which becomes infinitely large; or a change of sign for this determinant can enter only at such positions where becomes infinite.
In the differential equationy we have , and if , form a fundamental system of integrals of this differential equation, then
It follows that cannot become infinite or zero within the interval under consideration or upon the boundaries of this interval. Hence, it is again seen that cannot change sign within the interval .
If and are continuous within the interval , we have, through differentiating the equation , all higher derivatives of expressed in terms of and . Hence, if values of and are given for a definite value of , say , we have a power-series for (see Art. 79), which satisfies the equation .
Suppose that has a definite, positive or negative value for a definite value of situated within the interval , then on account of its continuity it will also be positive or negative for a certain neighborhood of , say . We may vary the curve in such a manner that within the interval it takes any form while without this region it remains unchanged.
Consequently the total variation, and therefore also the second variation of , depends only upon the variation within the region just mentioned, and in accordance with the remarks made above since we may find a function of the variable , which is continuous within the given region, which satisfies the differential equation , and which is of such a nature that and have given at values for , it follows that the transformation which was introduced is admissible, and we have
This quantity is evidently positive when is positive and negative when is negative, so long as
- (Art. 132).
We have then for the total variation
where denotes an expression of the third dimension in the quantities included within the brackets.
For small values of it is seen that has the same sign as the first term on the right-hand side of the above equation. We have, therefore, the following theorem :
The total variation the integral is positive when is positive, and negative when is negative throughout the whole interval .
If could change sign for any position within the interval , then there would be variations of the curve for which is positive and others for which is negative. Hence, for the existence of a maximum or a minimum of we have the following necessary condition :
In order that there exist a maximum or a minimum of the integral taken over the curve within the interval , it is necessary that have always the same sign within this interval; in the case of a maximum must be continuously negative, and in the case of a minimum this function must be continuously positive.
In this connection it is interesting to note a paper by Prof. W. F. Osgood in the Transactions of the American Mathematical Society, Vol. II, p. 273, entitled:
"On a fundamental property of a minimum in the Calculus of Variations and the proof of a theorem of Weierstrass's."
This paper, which is of great importance, may be much simplified.