CHAPTER V: THE VARIATION OF CURVES EXPRESSED ANALYTICALLY. THE FIRST VARIATION.
- 74 General forms of the variations hitherto employed.
- 75 The functions and . Their continuity.
- 76 Neighboring curves. The first variation.
- 77 The functions , and .
- 78 Proof of an important lemma.
- 79 The vanishing of the first variation and the differential equation .
- 80 The curvature expressed in terms of and .
- 81 The components and in the directions of the normal and the tangent.
- 82 Variations in the direction of the tangent and in the direction of the normal.
- 83 Discontinuities in the path of integration. Irregular curves.
- 84,85 Problem of Euler illustrating the preceding article.
- 86 Summary.
In Chapter II we considered examples of special variations. The method followed provided for the displacement of a curve in one direction only, in the direction parallel to the -axis, and is consequently applicable only to the comparison of integrals along curves obtained from one another by such a deformation.
We shall now give a more general form to the variations employed and shall seek strenuous methods for the solution of the general problem of variations proposed in Chapter I. After deriving the necessary conditions we shall then proceed to discuss the sufficient conditions. In order to develop the conditions for the appearance of a maximum or a minimum of the integral
it is necessary to study more closely the conception of the variation of a curve and fix this conception analytically.
By writing instead of each point , of a curve (presupposed regular) another point , , we transform the first curve into another regular curve. This second curve is neighboring the first curve if we make sufficiently small the quantities and which like and we consider as one-valued continuous functions of .
The following is one of the methods of effecting this result. Let and be continuous functions of and also of a quantity . We further suppose that and \eta vanish when for ever value of , for example
and being finite and continuous functions of .
The functions and and consequently also and are subject to further conditions. It is in general required to construct a curve between two given points which first may be regarded as fixed. Later the condition of their variability may be introduced. Consequently we have to consider only such values of that , and consequently , vanish on the limits.
If for , we write , , then for , we must write , . Further, the function must be developed in powers of , , and . For the convergence of this series, it is necessary that , , and have finite values.
Now, if we write
then we have
so that, whereas and have infinitely small values for infinitely small values of , the quantities , vacillate for between and and become infinite for . We shall consequently consider only such special variations in which and are functions of alone, and which with their derivatives are finite and continuous between the limits and . We thus restrict, in a great measure, the arbitrariness of the indefinitely small variations of the curve, and thus exclude a great many neighboring curves from the discussion. However, there exist among all the possible neighboring curves also such which satisfy the above conditions, and with these we shall first establish the necessary conditions and later show that the necessary conditions thus established are also sufficient for the establishment of the existence of a maximum or a minimum value of the integral. (See Arts. 134 et seq.)
We have instead of one neighboring curve a whole bundle of such curves if we make the substitutions
- , , , ,
and let , a quantity independent of the variables in , vary between and .
The total variation that is thereby introduced in the integral of the preceding article is
which developed by Maclaurin's Theorem is
Further (see Art. 25)
hence, equating coefficients of ,
So that 2) becomes
- , .
Owing to the hypotheses that , , , vary in a continuous manner with [that is, within the portion of curve considered no sudden change enters in the direction of the curve] , the first variation of the integral may be transformed in a remarkable manner.
and also (Art. 72)
and differentiating with respect to , we have
Writing , (Art. 73 ), and , and defining by the equation
it is seen that
In a similar manner it may be shown that
Lemma. If and are two continuous functions of between the limits and and if the integral
is always zero, in whatever manner is chosen, then necessarily must vanish for every value of between and .
The following proof, due to Prof. Schwarz, is a geometrical interpretation of a method due to Heine.
Suppose it possible that the function has a finite value for a point situated between and . Then owing to the continuity of we can find an interval within which also has a finite value.
We write the integral in the form
The second integral on the right hand side may be written
where is the mean value of for a value of within the interval .
We shall show that it is possible to determine a function which will render this integral positive and greater than the sum of the first and third integrals in the above expression, while at the same time .
Let us form the equation
which represents the parabola and the -axis.
Consider next the equation
where is a small quantity. By taking sufficiently small, this curve can be made to approach as near as we wish the parabola and the -axis.
Solving the above equation for , we have as the two roots (two branches)
is symmetrical with respect to the -axis, and for values of , such that , the ordinate of any point of the curve is greater than the corresponding ordinate of the parabola.
For the parabola , the integral
It follows then that for the curve we must have
for , we have ; and from the inequality
it follows for , that is positive and . For the lower branch is negative and the curve
follows the parabola to infinity as shown in the figure. It is however the upper branch which we use, since is less than , as soon as passes the value on either side of the origin.
Instead of the integral last written, take the integral which has the same value
where , and are mean values of in the respective intervals and where denotes that the quantity that stands within the brackets is less than .
It is seen that by taking sufficiently small that the sign of the integral is determined by that of and consequently this integral is different form zero.
Instead of the function , write
where and are positive integers.
We see that , and as above, it follows that
Hence, on the supposition that for a point of the curve alonjj which we integrate, it follows that a function can be found which causes the above integral to be different from zero.
But as this integral was supposed to be zero for all functions , it follows that we must have for all values of between and .
In the expression
unless <math\delta I</math> and always retain the same sign, it is necessary that be zero in order that be continuously negative or continuously positive; i.e., in order that the integral be a maximum or a minimum (see Art. 26).
Substitute for and their values in terms of from Art. 77, in the expression for of Art. 76, and we have
If we suppose that the points and are fixed so that for them, then the boundary terms vanish. Further, since is an arbitrary and continuous function of , it follows from the above lemma that, in order for to be zero, for every point of the curve within the interval . can not have a finite value different from zero for isolated points on the curve, since this portion of curve must be continuous in order that the integral may have a meaning.
The differential equation of the second order is a necessary condition for a maximum or a minimum value of , and will afford the required curve if such curve exists. We note that it is independent of the manner of variation, as the quantities and do not appear in it.
From the relation (see Art. 77)
it follows that
Among all possible variations there are those for which , and consequently
As above we have then
Further, if we multiply and respectively by and , we have by addition
and as and cannot both vanish simultaneously, it follows that . Hence the equations , on the one hand, and on the other, are necessary consequences of one another.
The equations often more convenient than ; especially is this the case if the function does not contain explicitly one of the two quantities and . If , for instance, is wanting, then , and from
it follows that constant.
The curvature at any point of a curve is denoted by
and owing to the equation
an expression which depends upon ,,, alone and not upon the higher derivatives.
It is thus seen that through the equation , a definite relation is expressed between the curvature of a curve at a point, the coordinates and the direction of the tangent of the curve at this point.
Let a point on the curve be transformed by a variation into the point and let the displacement be denoted by ; further, let the components in the and directions be and , while and denote the components of this displacement in the direction of the normal and the tangent to the curve at the point .
Let denote the angle between these directions and let the direction cosines of the normal be denoted by
- and ; i.e., by and .
Then from analytical geometry,
These expressions substituted in the formula for (Art. 79) give
From this it is seen that only the component of the variation which is in the direction of the normal enters under the integral sign.
By means of the formula below we will prove that the variation in the direction of the tangent brings forth only such terms for the first variation that are free from, the sign of integration.
As in Art. 76, write
and, substituting in this expression , , we have
it is seen that
so that everything under the sign of integration drops out, leaving
Hence, if we make a sliding of the curve by the substitution
and resolve this sliding into two components, of which the one is parallel to the direction of the tangent and the other is parallel to the direction of the normal, then the result of the sliding in the direction of the tangent is seen only in the terms which have reference to the limits, and all these terms are exact differentials under the sign of integration, while the effect due to a sliding in the direction of the normal is shown in the formula of the preceding article.
The expression for the first variation has been obtained on the hypothesis that the elements of integration have for each point of the path of integration a one-valued meaning. In case the path involved, discontinuities, it could be resolved into a finite number of portions of regular curve, and along each portion would have a meaning similar to that of the preceding article. It is assumed, then, that the required curve, which is to furnish a maximum or a minimum value of the integral, is regular in its whole trace, or, at least, that it consists of regular portions of curve. In the latter case we shall at first limit ourselves to the consideration of one such portion. Within this portion of curve not only , but also , will be one-valued functions of . This assumption is already implicitly contained in the assumption of the possibility of the development of by Taylor's Theorem; for otherwise the derivatives of according to and for the curve which has to be developed could not be formed.
That these assumptions have been made is due to the fact that otherwise the curve could not be the object of a mathematical investigation, since there are no methods of representing irregular curves in their entire generality. If therefore one is contented with the rules of differentiation and integration, he must extend these considerations over only such functions to which the rules may be applied, that is, to the functions having the properties above. There are many problems in geometry and mechanics for which the above hypotheses cannot be made.
The following problem proposed by Euler illustrates what has just been said:
Required a curve connecting two fixed points such that the area between the curve, its evolute and the radii of curvature at its extremities may be a minimum.
The analytical solution of this problem is the arc of a cycloid, if indeed there exists a minimum. We shall now show that such is not the case. For join the two fixed points and by a straight line which divide into equal parts, and draw alternately
above and below the line semi-circles having the parts of the line as diameters.
All the radii of curvature of each semi-circle, i.e., of each portion of curve which is to be a minimum, intersect on the line and it is evident that
must be a minimum.
If we increase the number sufficiently, we may make the above expression become arbitrarily small; and in the limit the curve will tend to become the straight line . From this it is evident that there is not present a minimum surface area.
The same result would have been obtained, if instead of the straight line we had taken the arc of a cycloid through these points, and had then drawn a system of small cycloids having their cusps along the large cycloid. (See Todhunter, Researches in the Calculus of Variations, p. 252.) The reason that a minimum is not given through the large cycloid is due to the fact that such a minimum is ofiEered by an irregular curve, and that this irregular curve is not included in our analytical research. It follows that our assumption made regarding the regularity of the curve is out of place and leads to something untrue.
But in spite of the not improbable possibility that the curve which is to satisfy a given proposition is irregular, we must make the hypothesis that the curve is regular, since we come to analytical difEerential equations only by limiting our investigations to such regular curves, and the most general theory of functions teaches that in turn through these diflEerential equations are defined the analytical functions which in their whole extent have existing derivatives.
To avoid any misunderstanding, we repeat what we have already said in the previous Chapter: it is not asserted that there is anything in the nature of the problem whereby one may a priori conclude that the required curve must be regular. Having these hypotheses, we fix our ideas and draw deductions. After the solution of the problem has been effected, we have to make in addition a special proof that the derived curve has all the required properties, and that this curve is the only one which has them.
The chief difficulty in all such problems, as we have shown above in the special problem of approximation (or of the passing to a limit), consists in showing that the regular curve that has been found, found indeed from the necessary conditions, also at the same time satisfies the sufficient conditions, and therefore satisfies all the requirements of the problem.
- ↑ Heine, Crelle, bd. 54, p. 338.
- ↑ See also Moigno et Liadelof , Calcul de Variations, p. 252.