CHAPTER X: THE CRITERIA THAT HAVE BEEN DERIVED UNDER THE ASSUMPTION OF CERTAIN SPECIAL VARIATIONS ARE ALSO SUFFICIENT FOR THE ESTABLISHMENT OF THE FORMULAE HITHERTO EMPLOYED.
- 134 The process employed is one of progressive exclusion.
- 135 Summary of the three necessary conditions that have been derived.
- 136,137 Special variations. The total variation.
- 138 A theorem in quadratic forms.
- 139 Establishment of the conditions that have been derived from the second variation.
- 140,141,142,143,144 Application to the first four problems of Chapter I.
The methods which we have followed would indicate that the whole process of the Calculus of Variations is a process of progressive exclusion. We first exclude curves for which is different from zero and limit ourselves to curves which satisfy the differential equation . From these latter curves we exclude all those along which does not retain the same sign. If, for any curves not yet excluded, at isolated points, we have simply a limiting case among those to which our conclusions apply. If for a stretch of curve not excluded by the above condition, we have to subject the curve to additional consideration in which the third and higher variations must be investigated. We further exclude all curves, in which conjugate points are found situated between the limits of integration, as being impossible generators of a maximum or a minimum. The cases in which no such pairs of points are to be found, or where such points are the limits of integration, require further investigation. This leads us to a fourth condition, a condition due to Weierstrass, which is discussed in Chapter XII. In this process of exclusion let us next see whether the variations admitted are sufficient for the general treatment under consideration.
As necessary conditions for the appearance of a maximum or a minimum, the following theorems have been established:
1) as functions of must be determined in such a – manner that they satisfy the differential equation .
2) Along the curve that has been so determined the function cannot be positive for a maximum nor can it be negative for a minimum; moreover, the case that at isolated points or along a certain stretch, cannot in its generality be treated, but the problems that thus arise m,ust be subjected to special investigation.
3) The integration may extend at most from a point to its conjugate point, but not beyond this point.
The last two conditions, which were derived from the consideration of the second variation, require certain limitations. On the one hand, a proof has to be established that the sign of is in reality the same as the sign of , if we choose for , , etc., the most general variations of all those special variations, for which the developments hitherto made were true ; it then remains to investigate whether and how far the criteria which have been established remain true for the case where the curve varies quite arbitrarily.
We return to the proof of the theorem proposed in the preceding article. We have, in the case of the investigations hitherto made, always assumed that , , , were sufficiently small quantities, since only under this assumption can we develop the right-hand side of
in powers of these quantities. This means not only that the curve which has been subjected to variation must lie indefinitely near the original curve, but also that the direction of the two curves can differ only a little from each other at corresponding points. We retain the same assumptions, and limit ourselves always to special variations.
We shall first prove that for all these variations the sign of and that of agree, so that for these variations the criteria already found are also sufficient. However, we no longer assume that the variations are expressible in the form , , where denotes a sufficiently small quantity.
Since the curvature of the original curve does not become infinitely large at any point (see Art. 95), and since further the original curve and the curve which has been subjected to variation deviate only a little from each other at corresponding points both in their position and the direction of their tangents, it follows that with each point of the original curve is associated the point of the curve that has been varied, in which the latter curve is cut by the normal drawn through the point on the first curve.
The equation of the normal at the point is
and from the remarks just made, the point , is to lie on this normal so that
If we consider this equation in connection with the definition of :
it follows that the variations may be represented in the form
In these expressions is an indefinitely small quantity, since and cannot both vanish at the same time (Art. 95), and it varies in a continuous manner with . Likewise the derivative of this quantity with respect to is an indefinitely small quantity which, however, may not be everywhere continuous.
Under the assumption that , , , are sufficiently small quantities, we may develop the total variation of the integral
with respect to the powers of , , , ; and, if we make use of Taylor's Theorem in the form
where , we have, since the terms of the first dimension vanish, a development of the form
If we further develop , etc., with respect to powers of , it is found that the aggregate of terms that do not contain is identical with which was obtained in Chapter VIII.
Integrating with respect to , we may represent the other terms as a quadratic form in , , , , whose coeflcients also contain these quantities and in such a way that they become indefinitely small with these quantities.
Next, writing in the values of , given in 1) and the following values of , also derived from 1):
where , , denote functions which still contain and , and in such a way that they become indefinitely small at the same time as these quantities.
After a known theorem in quadratic forms,
may always, through linear substitutions not involving imaginaries, be brought to the form
in such a way that at the same time the relation
is true, and where and are roots of the quadratic equation in :
Since the coefficients in this equation become simultaneously small with and , the same must also be true of and , the roots this equation.
If is the mean value between and , which also becomes indefinitely small with and , we may bring the expression
to the form
and consequently we have for the expression
and thus we have for the same form as we had before for (Art. 115).
We assume now that the necessary conditions for the existence of a maximum or a minimum are satisfied; that therefore along the whole curve , the function is different from zero or infinity, and always retains the same sign; that a function may be determined which satisfies the equation
and nowhere vanishes within the interval or upon the boundaries of this interval.
If we therefore understand by a positive quantity, and write
then the expression for above becomes
If is given a fixed value, then we may choose , so small that the absolute value of the quantity that depends upon them is less than . The quantity is therefore positive, and consequently also the second integral of the above expression. We have yet to show that the first integral is also positive, if .
After a known theorem in differential equations it is always possible, as soon as the equation
is integrated through a continuous function of , which within and on the boundaries of the interval nowhere vanishes, also to integrate the differential equation
through a continuous function of , which, if does not exceed certain limits, deviates indefinitely little throughout its whole trace from , and may therefore be represented in the form
where becomes indefinitely small at the same time as for all values of that come into consideration.
The function will therefore vanish nowhere within the interval . In this manner a certain limit has also been established for , which it cannot exceed ; but if the condition is also added that must be so small that has the same sign as , then , may always be chosen so small that .
The first integral may then be transformed in a manner similar to that in which the integral 8) of Art. 115 was transformed into 14) of Art. 119, and we thus have
which shows that for all indefinitely small variations of the curve which have been brought about under the given assumptions, is positive if is positive. If is negative, the same determinations regarding remain ; only must be chosen negative and . Both integrals on the right of the above equation are then negative, and consequently is itself negative.
We have therefore proved the assertion made above : If in the interval the necessary conditions which were derived from the consideration of the second variation of the integral for the existence of a maximum or a m-inimum, are satisfied, then the sign of the total variation will be the same as the sign of the second variation for all variations of the curve which have been so chosen, that not only the distances between corresponding points on the original curve and the curve subjected to variation are arbitrarily small, but also the directions of both curves at corresponding points deviate from, each other by an arbitrarily small quantity.
It has thus been shown that the three conditions given in Art. 135 are necessary for the existence of a maximum or a minimum. A further examination will give a fourth condition (Weierstrass's condition, see Chapter XII) whose fulfillment is also sufficient. This condition, if fulfilled, is then decisive, after we have first assured ourselves that the other three conditions are satisfied.
APPLICATION OF THE ESTABLISHED CRITERIA TO THE PROBLEMS I, II, III AND IV, WHICH WERE PROPOSED IN CHAPTER I AND FURTHER DISCUSSED IN CHAPTER VII.
Problem I. The problem of the minimal surface of rotation.
As the solution of the equation , we found (Art. 100) the two simultaneous equations of the catenary :
We have, therefore (Art. 125),
- and consequently,
If, now, are the values of which correspond to the value , then is
- [cf. 2)],
In order to find the point conjugate to we have to write in this expression for their values in terms of and then solve the equation .
To avoid this somewhat complicated calculation, however, we may make use of a geometrical interpretation (Art. 58). The equation of the tangent to the catenary at the point is
Therefore, the tangent cuts the -axis in the point determined through the equation
The tangent at any point of the catenary cuts the -axis at a point determined by the equation
If, now, the point is to be conjugate to , then its coordinates must satisfy 4), which becomes
Hence, since and do not vanish (Art. 101), we have
that is, the conjugate points of the catenary have the property that the tangents drawn through them cut each other on the -axis. We thus have an easy geometrical method of determining the point conjugate to any point on the catenary.
Further we have
and since is always positive, and cannot simultaneously vanish, it follows that is always positive and different from zero and infinity. Hence, the portion of a catenary that is situated between two conjugate points, when rotated about the -axis, generates a surface of smallest area (cf. Art. 167).
At the same time in this problem it is seen how small a role the condition regarding , has played in the strenuous proof relative to the existence of a minimum.
Problem II. Problem of the brachistochrone.
In this problem the expression for is found to be
We assumed from certain a priori reasons that between the points and of the curve there could be present no cusp (see also Art. 104); that is no point for which and are both equal to zero simultaneously. For such an arc of the curve is then always positive and different from zero and infinity, since the quantities under the square root sign are always finite and different from zero (see also Art. 95).
We obtained (Art. 103) the solution of the equation in the form
where here is written in the place of , and in the place of , and instead of ; is a given quantity which is determined through the initial velocity.
We consequently have
With the positions which we have assumed for and both and are different from and , and consequently the equation for the determination of the point conjugate to has the form
which is a transcendental equation for the determination of .
We easily see that there is no other real root within the interval except , since the derivative of , namely, is negative, so that continuously decreases, if deviates from , and can never again take the value .
Consequently there is no point conjugate to the point on the arc of the cycloid upon which lies, and therefore every arc of the cycloid situated between two cusps of this curve has the property that a material point which slides along it from a point reaches another point of the curve in the shortest time (Art. 168).
In this problem we see that the condition was sufficient to establish the existence of a minimum. The case where the initial velocity is zero and the point is situated at one of the cusps will be discussed later (Art. 169).
Problem III. Problem of the shortest line on the surface of a sphere.
In this problem we find that
This expression cannot become infinitely large, since and cannot simultaneously vanish.
However, the function , will vanish if ; that is, when or . Consequently, in this case, we must so choose the system of coordinates that nowhere along the trace of the curve becomes equal to zero or to . If this has been done, then for the whole stretch from to is positive, and does not become zero or infinitely large.
The equation furnishes the arc of a great circle, whose equations are (see Art. 106):
Accordingly, we have
Hence, since for the point we have , it follows that
Therefore, in order to find the point conjugate to the point , we have to solve the equation with respect to .
Since the denominator of 3) cannot become infinite, the conjugate point is to be determined from the equation . We consequently have as the point conjugate to ; that is, the point conjugate to is the other end of the diameter of the circle drawn through .
Hence the arc of a great circle through the points and , measured in a direction fixed as positive, is the shortest distance upon the surface of the sphere only when these points are not at a distance of or more from each other, a result which is of itself geometrically clear.
We may remark that the condition that cannot vanish is clearly in this case unnecessary ; since the arc of a great circle possesses the property of a minimum independently of the choice of the system of coordinates with respect to which , say, at some point of the curve vanishes.
From the figure in Art. 107 it is clear that when is the pole of the sphere, the family of curves passing through and satisfying the differential equation (i.e., arcs of great circles) intersect again only at the other pole. In the next Chapter it will appear that the two poles are conjugate points. This, together with what was given in the preceding article, may be taken as a proof that the arcs of great circles can meet only at opposite poles.
Problem IV. Problem of the surface offering the least resistance.
In this problem let us write (Art. 110)
Now the tangent to the curve at any point is
and the intercept on the -axis is
The tangent to the curve at any point cuts the -axis where
We therefore have for the determination of the point conjugate to the equation
- or .
As in Art. 140, this gives an easy geometrical construction for conjugate points.
- ↑ Such substitutions are called by Cayley orthogonal (Crelle, bd. 32, p. 119); see also Euler, Nov. Comm. Petrop., IS, p. 275; 20, p. 217; Cauchy, Exerc. de Math., 4, p. 140; Jacobi, Crelle, bd. 12, p. 7; bd. 30, p. 46; Baltzer, Theoiie und Anwendungen der Determinanten, 1881, p. 187; Rodrigves, Liouv. Journ., t. S, p. 405; Hesse, Crelle, bd. 57, p. 175.