Calculus of Variations/CHAPTER XV
CHAPTER XV: RESTRICTED VARIATIONS. THE THEOREMS OF STEINER.
- 197 Variations along two different portions of curve.
- 198 Variation where a point must remain upon a fixed curve.
- 199 Application to a particular case.
- 200 Variation where a part of the curve coincides with a fixed curve.
- 201 Generalizations involving several variables and several integrals.
- 202 The isoperimetrical problem when the circle (Art. 195) which incloses the given area cannot be inscribed within fixed boundaries.
- 203 Statement of two problems due to Steiner. Criticism of his assertion that the Calculus of Variations was not sufficient for the proof of these problems.
- 204 Two problems due to Weierstrass which are more general than Steiner's and their proof by means of the Calculus of Variations.
- 205 The behavior of the -function at fixed boundaries.
- 206 Further discussion regarding this function.
- 207 The case that there is a sudden change in the direction of the boundary curve at the point where it is approached by the curve that is free to vary.
- 208 The case where the curve meets the boundary at a point and then leaves it.
- 209 The tangents to the two portions of curve make equal angles with the tangent to the fixed curve.
- 210 The isoperimetrical problem reversed.
- 211 Consideration of the problem: Three points not lying in the same straight line are given in the plane. It is required to draw a line through them in a definite order which with a given length includes the greatest possible surface area.
- 212 Expression for portions of the curve that overlap.
- 213 The solution of the differential equation may be straight lines or arcs of circles.
- 214 The problem reduced to a problem in the Theory of Maxima and Minima.
- 215 The problem solved.
We shall consider in this Chapter some special cases of restricted variations. Suppose first that the path of integration is taken over two traces , and . We have for the first variation of the integral (Art. 79)
Since the variation along the traces and is free, it follows that for them, and consequently
In order then for the first variation to be zero, it is necessary that
If first the conditions of the problem leave free to vary in any direction, we must have, since and are arbitrary,
or, the curve consists of a single trace
Secondly, if the conditions of the problem require , to remain upon a fixed curve , then since the displacement is in the direction of the tangent to this curve, the expression
may be replaced by
where is the angle betvsreen the fixed curve and the -axis
We may apply the above results to the function
In the first case, where the point can move at pleasure, we have
so that, unless vanishes at , we must have
where and are the angles that the tangents to the variable curve at the point make with the -axis. From this it follows that and are not different traces but constitute a single curve with one tangent at the point . In the second case, where is constrained to lie upon the fixed curve (see Fig. in the preceding article), we have
From this it follows, unless at the point , that
It is seen that the tangents to the two traces and at the point have either one and the same tangent at and are parts of one and the same curve, so that this case is the same as if were not constrained, or they make with the tangent to the fixed curve equal angles and .
A limiting case is where , when again and form a continuous curve touching the fixed curve at the point . The function is here
- , when
Suppose that the path of integration coincides in part with one or more fixed curves, for example, with the curve :.
Then we cannot say that for the path of integration from to , but from the expression
it is evident that for the possibility of a maximum', and must have opposite signs, and the same signs for the possibility of a minimum.
In the general case, we made the substitutions
Suppose that some point of the path of integration is constrained to remain on a fixed curve, and, for simplicity, suppose that
but and at .
Our previous equations (Art. 184) become now
As in our previous discussion (Art. 184), it follows that
If we write
and denote by , the angle which the tangent to the fixed curve at the point makes with the -axis, the above expression becomes
If the point were not restricted, then and would be arbitrary, and we would have here
which results compare with those of Art. 199.
We saw in the previous Chapter, if there existed a closed curve which with a given length bounded a maximum surface area, that this curve was a circle. We supposed that it was possible for the circle to be situated entirely within the boundary of a given region. Suppose that this is not the case. The curve must then at least touch the given boundaries in two points or have a portion of the boundary in common. For we saw that the curve consisted of arcs of equal radii, and if these arcs did not touch the boundaries, there would necessarily be discontinuous changes in the direction of the variable curve. At such places, however, the surface-area could be increased without changing the perimeter.
Regarding the nature of the curve when it touches the boundaries, Steiner has given the two following theorems :
1 ) If the curve coincides with a portion of the boundary , then the free portions of this curve are arcs of circles of equal radii, which are tangent to the boundary at the points of contact.
2) If the curve touches the boundary of the region in a point, then both parts of the curve are arcs of circles of equal radii, and the tangents to these two arcs at the point of contact with the boundary make with the tangent to the boundary at this point, equal angles.
Steiner proved these thorems in a synthetic manner, and remarked that a synthetic-geometrical treatment seemed necessary, because the principles of the Calculus of Variations were not sufficient. Such remarks were, in a measure, justifiable, since up to that time only curves had been considered which satisfied the differential equation throughout their whole extent, and, therefore, no analytical means were known for the treatment of curves which in part coincided with given curves. However, there was no reason for saying that a method for the treatment of such problems was not within the province of the Calculus of Variations.
We shall show that the principles of the Calculus of Variations are sufficient to establish Steiner's theorems by proving two theorems due to Weierstrass, which are more general than the theorems of Steiner, and which have reference to the behavior of a curve at the points where it touches the boundary. The two theorems of Steiner are special cases of these theorems.
Suppose that the curve which satisfies the differential equation approaches the boundary at the point 1 and coincides with it up to the point 2. On the part of the curve which is traversed before we come to the boundary at 1, we take a point so near to 1 that between and 1 there is no sudden change in the direction of the curve.
The portion of curve 012 shall be so varied that we come to the boundary along another path from to a point 3 before 1 or from to a point 4 after 1 and then traverse the boundary to 2.
As we have already seen (Art. 161) the variation thereby produced in the integrals and may be expressed as follows: Let , 1 be the direction-cosines of the curve 01 at the point 1; the direction-cosines of the boundary at this point; the coordinates of the point 1, and the element of length of the boundary. Then we have, if the boundary is approached before the point 1 [see formula 5) Art. 161]
and if the boundary is approached after the point 1 [see formula 6) Art. 161]
Hence for case 1): ,
for case 2:
If the curve is to cause to have a maximum or a minimum value while remains unchanged, then (cf. Art. 189) must have the same sign for both of the above variations. Hence, if the curve satisfies the differential equation , and if we write
then the function must be zero at the point 1 of the boundary, because otherwise we could choose so small that the at at sign of the whole expression depended upon the sign of the linear term, which in the first case is positive and in the second negative.
We saw (Art. 157) that
If is different from 0, (which must be determined in each separate case), it follows that
We wrote (Art. 157)
and consequently if we take the lower sign, so that then it may happen that becomes infinitely large within the limits of integration, because for the value both and are zero (see Art. 157).
In general, we have
- (cf. Art. 199).
A special investigation must be made in the other case for every particular problem. We, therefore, have the theorem :
If the curve which satisfies the differential eguation approaches the boundary at a point and then coincides with a portion of the boundary , the direction at the point of contact can suffer no discontinuous change.
The same result is derived in an analogous manner for the point where the curve leaves the boundary after having coincided with a portion of it.
We have tacitly assumed that there is no sudden change in the direction of the boundary at the point 1. But if this is the case and if are the direction-cosines with which one approaches the point 1, and those with which one leaves the point 1, then we have for the expression:
in the first case:
in the second case :
In a following Chapter (Art. 221), it will be proved that, if a maximum or a minimum is to appear, the function must have continuously the same sign for every point of the curve which is varied and for arbitrary directions along it; in the first case this sign must not be positive, and in the second case it must not be negative.
From this it follows, for the case of both maximum and minimum, that we must again have
while remains arbitrary.
For, if we are seeking a minimum, after the theorem just cited, the function cannot be negative; but it cannot be positive because in virtue of the equation
would for certain variations experience a negative change. Hence we must have:
The same is true of the point where the curve leaves the boundary so that we have the same results for the end-point as those just given for the initial-point. The results may be stated as follows:
If the curve for which there is to appear a maximum or a minimum meets the boundary and traverses a portion of it, then at the point where it first comes to the boundary and at the point where it leaves the boundary , the two curves must be so situated that the tangents are the same for both curves. But if at these points there is a discontinuous change in direction of the boundary curve, then the direction of the curve as it approaches the boundary and that of the boundary at the point of approach may be quite arbitrary.
This is the first of Weierstrass' theorems.
We consider next the case where the curve meets the boundary in one point and then leaves it. Let 01 and 12 be the two portions of curve that satisfy the differential equation and meet the boundary at the point 1. Take the points 0 and 2 so near to 1 that within the intervals 01 and 12 there are no sudden changes in direction. We vary the curve 012 by going from the point 1 to a point 3 on the boundary. The point 3 is connected with the points and 2 by curves which do not necessarily satisfy the differential equation, but are subject to the condition that the integral remains unaltered by this variation.
Let be the direction of 01 at 1,
the direction of 12 at 1,
and let the coordinates which belong to the diferent points be indicated by the corresponding indices.
Then, as we have already seen (Arts. 79 and 154)
If we assume that is different from zero, and consequently that the tangents to the two portions of curve 01 and 12 at the point 1 do not coincide, then we may write
The geometrical meaning of and is seen, if we consider that in virtue of the two above relations the length 13 is the geometrical sum of the two lengths and and that consequently and are the coordinates of the line 13 with respect to an oblique system of coordinates whose positive axes have the directions, and , and are consequently represented by the tangents of the two portions of curve at the point 1. If we write these values for in the above expressions, we have
The straight line whose equation is divides the plane into two halves ; for the points of one-half, , and for the other half, . The point 1 which the curve has in common with the boundary can move only along the boundary. If the direction of the tangent to the boundary at the point 1 was different from the direction of the straight line , which may be called the dividing line, then by sliding the point 1 in opposite directions the quantity would be either positive or negative ; and since this quantity (neglecting a constant factor) is the distance from the dividing line, it is seen that it becomes infinitely small of the first order with so small that
has the same sign as , and, therefore, may be either positive or negative. Hence we must have
Accordingly, the direction of the tangent to the boundary curve must coincide with that of the dividing line.
The sines of the angles which the dividing line makes with the and -axes, that is, with the tangents to the two portions of curve at the point where they meet the boundary, are to each other as is to , if the two angles are measured in opposite directions.
Weierstrass' second theorem may accordingly be stated as follows :
If the curve which satisfies the differential equation meets the boundary in only one point and then leaves it, the tangents to the two portions of curve at this point, make with the tangent to the boundary at the same point, angles whose sines are to each other as is to .
The second theorem of Steiner relative to the isoperimetrical problem is only a special case of this theorem. In this problem we have so that the two angles which the two tangents to to the curves make with the tangent to the boundary curve are equal.
We have shown that the curve in every point where the variation is free satisfies one and the same differential equation and that the constant has the same value for the whole curve (Art. 185). This leads to a certain paradox: If we reverse the isoperimetrical problem and seek the shortest line among all those lines which inclose a given surface area, we come to the differential eqation of the isoperimetrical problem.
We have in the place of which occurred before; still on this account the nature of the differential equation is not changed, since there is only a change in the constants. It is, however, a priori clear that the solution of the two problems must be the same; for, if it were possible to keep the surface area constant and shorten the perimeter, it is evident that with the original perimeter we could have inclosed a greater surface area. Hence, the curve, which has been derived from the differential equation of the first problem, satisfies also the inverse problem. We consequently have as the solution of the second problem the theorem: The curve, wherever there is free variation, consists of arcs of circles which have equal radii.
Problem. Three points 1, 2, 3 not lying in the same straight line are given in the plane and it is required to draw a line through them in a definite order, which includes a given surface area and at the same time has the shortest possible length.
We know that a circle , say, fulfills these requirements, if the given area is the same as that included by a circle, which is determined by the three points 1, 2, 3.
But if the surface area is greater or smaller than , then the arcs of circles must be drawn outward or inward. If, however the area is very small, we cannot draw arcs of circles so as to inclose this area without crossing one another, and we do not admit into consideration the areas that are described in the opposite directions.
The problem may be solved as follows : The curve, although not being limited by further conditions, need not vary everywhere in a free manner, and, consequently, it is not necessarily constituted out of arcs of circles. For if we assume that the curve is not to cross itself, then of itself it may offer barriers which obstruct free variation.
If, for example, the curve 0 1 2 3 partially overlaps so that the portion 1 3 coincides up to the point 2 with the portion 01, then among all possible variations, there are present those where 01 remain unchanged and only 1 3 varies ; and since the curve is not to cross itself, the variation of the portion 1 2 can take place only on the side of 1 on which the point 3 lies, and, consequently the freedom of the variation of the curve is essentially limited.
In itself the requirement that the curve is not to cut itself is not necessary, as the integrals that appear have a meaning also for this case.
If there are overlapping portions of curve, then we may allow such variations to enter that points coincident before the variation may also coincide after the variation, without the second integral changing its value. We shall investigate the kind of differential equation that is thereby produced for these portions of curve.
The following investigation is also applicable to the case where the second integral is not present. We have simply to make .
We introduce the variations
It has been shown that the first variation of is identical with
provided that can be expressed as a power-series in in such a way that the total variation of the second integral vanishes.
This can be brought to the form
In the former treatment and were entirely arbitrary, except that at certain points and along certain portions of curve they vanished. Wherever they were arbitrary it was necessary that . In the case before us we have in addition those portions of curve which overlap the curve several times without crossing it. The differential equation, which these portions of curve satisfy, may be obtained as follows:
We have, since is a positive increment of (Art. 68),
Let 1 2 be a portion of curve that is traversed several times. The integral over this portion of curve, after it has been traversed once from the point 1 to the point 2, may be written in the form
The portion of the integral taken over the curve in the opposite direction is
If this portion of curve is traversed times in the first direction and times in the second, and if all the variations except those that relate to this portion of curve be put equal to zero, then the variation of the whole integral is equal to the variation of the sum of integrals:
the above sum is equal to
or, if we put
the sum is
The portion of curve 1 2 is traversed only once for this integral, and consequently the variations are quite free. The interval 1 2 must therefore satisfy the differential equation which is derived for the function in the same manner as in the former investigations, where was the function considered.
If, for example, the problem is to determine the curve which with a given surface area has the shortest perimeter, then
and for ,
Consequently the differential equation leads to a straight line.
But if , we have
The corresponding differential equation is of the form
where ; it, therefore, leads to the arc of a circle which has a different radius than the one belonging to the portions of curve where the variation is free.
This case, however, does not in reality appear unless there are certain modifications ; for, if we traverse such an arc of circle twice in opposite directions, the portion of surface area thereby obtained is zero. We may, however, shorten the perimeter by taking instead of the arc of a circle the chord which joins its endpoints, this being the first solution above. If, further, the same arc of circle was traversed several times, then in case there are not special modifications, we may neglect the first two times or the first times that the arc is traversed (owing to which the perimeter is shortened) without changing the surface area,
Taking also into consideration the case where , when a straight line enters, we have to see which of these portions of curve (straight line or arc) can be used to form the required curve and how they are to be grouped. We have then to seek all possible kinds of combinations and make proof of their admissibility.
We consider any configuration and cause it to vary. Since the nature of the curve is known and only the end-points of the individual portions are undetermined, we have to subject these to variations. The previous theorems are fully sufficient for carrying this out. We, therefore, have a means of determining whether such a configuration of the individual portions is, or is not possible.
Since the individual portions satisfy their differential equations, the first variations of the corresponding integrals will depend only upon the variation of the end-points; and, if we apply this to all the portions of the curve, we will have a linear function of all the variations of the coordinates of the individual end-points.
These end-points may be subjected to further restrictions; for example, they may be compelled to lie upon given curves, etc.
By the application of previously developed theorems, we have certain equations for the determination of the possible position of the end -points of the individual portions and we may thus see whether a definite configuration is, or is not possible.
At all events, for the grouping which has been thus determined the first variation of the integral vanishes, but this does not of itself denote that a maximum or a minimum has appeared. This determination is a problem in the usual Theory of Maxima and Minima. Since, as soon as the individual portions of curve have been found, we can also determine the integrals for them whose values depend only upon the constants that have been introduced and the coordinates of the end-points. We have thus an ordinary function of a finite number of variables, and the question is whether this function really satisfies the conditions of a maximum or a minimum. This subject is treated in the Theory of Maxima and Minima, involving several variables.
Thus we may at least determine whether or not a certain formation of the curve satisfies the problem. For example, a curve is required to pass in a definite order through the points 1, 2 and 3 and which having the smallest possible perimeter is to inscribe a given surface area. The curve in question consists of three portions which pass through 1 and 2, 2 and 3, 3 and 1. These portions are the arcs of circles with equal radii, if the given surface area is sufficiently large. This radius is to be determined from the given value of the surface.
The integral is a function of the constants that appear, and it may be shown that this integral is in reality a minimum when the constants have been correctly determined.
But if the surface area is not sufficiently large, then the portions of curve must partially overlap one another, and the portions along which this happens are straight lines. The curve cannot end in points which are perfectly free to vary; for if this were the case, we could so vary the point that the surface area remained the same while its length became shorter. These points must lie along straight lines which pass through the three given points.
It is thus found that the curve consists in reality of three arcs of circles which are described with equal radii and which mutually touch one another and go off into straight lines that pass through the given points, as shown in the figure.
It is seen that the solution of the problem is independent of the position of the points 1, 2, 3 relative to one another; for we can slide the points 1, 2, 3 backward and forward upon the straight lines without causing the curve to lose the property of having the minimum length. It is essential only in what manner the points are chosen where the straight lines come together with the arcs of the circles. These points corresponding to the points 1, 2, 3 may be denoted by 1', 2', 3'. If the portion 2' 1' 1 be considered as a fixed boundary and the end-point of 3' 1' varies along it, it follows from a theorem already given ( Art. 206 ), that 3'1' must so touch the boundary, that the curve 3' 1' 1 does not change its direction abruptly. Hence every two arcs of circles must touch at the points where they come together. Since the radii of the arcs of circles are equal, it follows that the three centers of the arcs of circles form an equilateral triangle, and consequently the three arcs of circles are of equal length. Therefore every two straight lines form an angle of with each other, and thus the solution of the problem is uniquely determined. The above problem was proposed by Todhunter in the Mathematical Tripos Examination of 1865. It is treated by him (Researches in Calculus of Variations, pp. 44 et seq. ).