Calculus of Variations/CHAPTER I

CHAPTER I: PRESENTATION OF THE PRINCIPAL PROBLEMS OF THE CALCULUS OF VARIATIONS.

  • 1 The connection between the Calculus of Variations and the Theory of Maxima and Minima.
Problem I. The curve which generates a minimal surface area when rotated about a given axis.
  • 2,3,4 The solution of this problem by the methods of Maxima and Minima.
  • 5,6 The difference between the Calculus of Variations and the Theory of Maxima and Minima.
  • 7 The coordinates , expressed as functions of a parameter . Problem I formulated in terms of the parameter .
  • 8,9 Problem II. The brachistochrone.
  • 10 Problem III. The shortest line on a given surface.
  • 11 The advantage of formulating the problems in terms of the parameter .
  • 12 Problem IV. The surface of revolution of least resistance.
  • 13 The general problem stated.
  • 14 The variation of the endpoints.
  • 15 Problem V. The isoperimetrical problem.
  • 16 Problem VI. The curve whose center of gravity lies lowest.
  • 17 Statement of the general problem in Relative Maxima and Minima.
  • 18 Generalizations that may be made.
  • 19 Variation of a curve. Analytical definition of maximum and minimum. Neighboring curves.
  • 20 A different statement of the general problem.
  • 21 Inadmissibility of the presupposed existence of a maximum or a minimum.
  • Problems

Article 1.
At the time when the Differential Calculus, and in part also the Integral Calculus, were being formulated, certain problems were proposed, which, although not belonging to the province of the Theory of Maxima and Minima, had a marked semblance to the problems of that theory, and were often solvable by methods belonging to it. The following was one of the first problems proposed:

Problem I. Two points and with coordinates and respectively are given. Both points lie on the same side of the axis of in the plane-. It is required to join and by a curve which lies in the upper half of the -plane (axis of inclusive) such that when the plane is turned through one complete revolution about the axis of , the zone generated by this curve may have the smallest possible surface area.

We may use this problem to illustrate the connection between the Calculus of Variations and the Theory of Maxima and Minima; at the same time the difference between the two theories is evident.

Article 2.
If we try to solve the problem of the preceding article by the methods of the Theory of Maxima and Minima, we must proceed as follows:

Suppose that it is possible to draw a curve between and which satisfies the problem. Then every portion of this curve, however small, must have the property of generating a surface of smallest area. For, suppose a change is made in an arbitrary portion of the curve, however small, and let the remaining portion of the curve be unchanged. If by this change the surface area generated by this arbitrary portion of curve is less than it was before, then the curve containing the deformed portion of curve generates a smaller surface area than the original curve. Also, if to one value of there belong several values of , then instead of the portion of curve belonging to the same abscissa, we might take the straight line which joins these two points. This line would generate a surface of smaller area than that generated by the curve that passes through the same two points. Hence the curve would generate a surface which did not have a minimum area. We may therefore consider the curve as divided into portions such that the projections of these portions on the axis of are all equal.

Calculus of Variations Harris Hancock Article 2 graphic.png

Article 3.
The above hypotheses being granted, we suppose that the two points and are taken on the curve, and we find another point on the curve such that . We suppose that and , and are joined together by straight lines, and later we suppose that these straight lines are taken so close together that there is a transition from the straight lines to the curve. The remaining portions of curve on the left-hand side of and on the right-hand side of are supposed to remain unaltered.

Calculus of Variations Harris Hancock Article 3 graphic.png

The portions of surface area generated by the straight lines and are

and

In order to have a minimum the sum of these two expressions when differentiated with regard to y must be zero; i.e.,

The quantity may be determined from this equation as a function of , so that , say. We therefore have and . Hence by Taylor's Theorem,

and consequently,

Substituting these values in [A], we have, neglecting the factor ,

Expand this expression in ascending powers of , divide through by and then make . We then have

;

or

Therefore in order to have a minimum value, or must satisfy this differential equation; however, when satisfies this differential equation we do not always have a minimum, as will be shown later.

In other words, the differential equation [B] is a necessary consequence of the supposed existence of a minimal surface of revolution. As a condition, however, it is not sufficient to assure the existence of a curve giving such a surface.

Differentiate the equation [B] with regard to , and we have

,

or

.

Integrating, we have

where is the constant of integration. Since and are two solutions of this last differential equation, the general solution is

,

where and are constants. This last equation is that of the catenary curve.

Article 4.
Thus, by the help of the Theory of Maxima and Minima, we have, it is true, come to a certain result; but, on the other hand, we have yet to ask whether this curve gives a true minimum; and owing to the manner in which we have arrived at these conclusions, we have yet to see whether this curve only in a definite portion or throughout its whole extent possesses the property required in the problem.

That we are justified in insisting upon this last statement is seen from what follows later, where it will be shown that the curve found above satisfies the required conditions only between given limits.

A simple consideration shows that the method we have followed above is not at all rigorous; since it presupposes, which of itself is not admissible, that the curve which satisfies the problems is regular in its whole extent, for otherwise the portions of curve between the two points and could not be replaced by straight lines joining these two points; also, the expansion by Taylor's Theorem would not have been admissible.

Article 5.
The characteristic difference between problems relative to Maxima and Minima and the problems which have to do with the Calculus of Variations consists in the fact that, in the first case, we have to deal with only a finite number of discrete points, while in the Calculus of Variations, the question is concerning a continuous series of points.

If we wish to substitute in the place of the curve first a polygonal line and afterwards apply to this line methods similar to those used above, then it turns out that, after we have found a line which satisfies all the conditions, it is necessary yet to prove that the required limiting transition from polygonal line to curve in reality results in a definite curve which satisfies the conditions of the problem.

Article 6.
Every limiting transition, as from polygon to curve, is made of itself, if we make use of the conception of integration, since an integral represents the limiting value of a sum of quantities which, following a definite law, increase so as to become infinite in number, the quantities themselves becoming smaller in a corresponding manner.

If we therefore define the surface area of the curve , which we have to find, by

,

or

,

then this integral will have a definite value for every curve that is drawn between and , and consequently the problem may be stated as follows:

Problem I. is to be so determined as a function of that the above integral shall have the smallest possible value.

The solution of this problem will be given later. The two methods given above have been chosen to make clear what there is in common in the Theory of Maxima and Minima and the Calculus of Variations, and also to show the difference between them.

In the Differential Calculus a definite function is given, and a special value of the variable or variables (if there are more than one variable) is sought, for which the function takes the greatest or least possible value; in the Calculus of Variation a function is sought and an expression is given which depends upon this function in a certain known manner. A definite integral is considered, in which the integrand depends upon the unknown function in a known manner, and it is asked what form must the unknown function have in order that the definite integral may have a maximum or a minimum value.

We treat only real values of the variables.

Article 7.
If and the point corresponds to and to , then with reference to is known as a later point; and with reference to is known as an earlier point.

Calculus of Variations Harris Hancock Article 7 graphic.png

As was shown in Art. 2, the ordinate of the required curve is a one-valued function of the abscissa . It often happens that one cannot know à priori that one of the ordinates is a one-valued function of the other. Poincaré[1] has shown that it is always possible to express the two variables , , when there is an analytic relation between them, as one-valued functions of a third variable . The only property that is required of this variable is, when it traverses all values between two given limits, the corresponding point traverses the curve from the initial point to the end point, and in such a way that for a greater value of there belongs a later point of the curve.

For example, suppose that where and are two independent variables. Then in virtue of this equation there is no way of expressing the dependence of one of these variables upon the other without the introduction of transcendental functions. But if we write

then

or

Thus and are one-valued functions of the variable .

If then we introduce such a new variable in the integral of Problem I, that integral becomes

,

where we denote by and the quantities and .

We may now state Problem I as follows:

The quantities and are to be determined as one-valued functions of a parameter in such a way that the above integral will have the smallest possible value.

Article 8.
That we may learn the essential properties of the Calculus of Variations, we shall next formulate other simple problems; then, while we seek the general characteristics of these problems, we shall of our own accord come to a more exact statement of the problems which the Calculus of Variations has to solve.

As a second problem may be given the very celebrated problem of the Calculus of Variations, that of the brachistochrone[2] (curve of quickest descent), which may be stated as follows:

Problem II. Two points and are situated in a vertical plane, the point being situated lower than the point ; a curve is to be drawn between these points in such a manner that a material point subject to the action of gravity and compelled to move upon this curve with a given initial velocity, shall go from the point to the point in the shortest possible time.

Let the mass of the material point be 1, its initial velocity the acceleration of gravity , the time , and the coordinates of and respectively and . Let the direction of the positive -axis be the direction of a falling body (due to gravity) and let the positive -axis be directed toward the side on which the point lies. Then, according to the law of the Conservation of Energy,

,

or

;

whence

.

We have then as our problem: so determine as a function of that the above integral shall have the smallest possible value.

As regards the signs of the roots that appear in the above integral, it is evident that these signs must be the same at the beginning of the motion and may be taken positive. For on mechanical grounds it follows that the curve must at first descend; consequently at the beginning of the motion increases with increasing , and is therefore positive. Since always a positive quantity, being equal to , and can never vanish, we may always give to the positive sign. Also at the beginning of the motion the quantity must have the positive sign, since always represents a positive increment of time. However, in the further course of the motion, it may happen that . Then the quantity passes through infinity, so that and may simultaneously change their sign, while continues with the positive sign.

Article 9.
The assumption made in the statement of the problem that must lie below is not essential. For the material point has at a certain velocity , which we may calculate from the initial velocity and the height of above . When the point reaches with this velocity it may rise again, and it will have the original velocity when it has reached the height on the other side of . The time which is necessary for the ascent is the same as that required in the descent, if we assume that the curve along which the ascent takes place is symmetrical with that of the descent.

If, therefore, the point started from , we could calculate from , which is now the initial velocity, the velocity at the point . We then have the curve in question, if we seek the curve along which the point with the initial velocity reaches in the shortest time.

In the case of the present problem we see from physical considerations that is a one-valued function of . As this is not possible in all cases, it is expedient to represent the curve here also by two equations; that is, to consider and as one-valued functions of a third variable [3] where is subject to the only condition, that when it goes through all values between two given limits, the corresponding point , traverses the curve from the beginning-point to the end-point and in such a way that to a greater value of there corresponds a later point of the curve.

The above integral becomes

,

where we have written and for and respectively.

Our problem then is: Determine and as functions of a parameter in such a way that the integral just written may have the smallest possible value.

Article 10.
Problem III. Between two points on a regular surface , a curve is to be drawn so that its length is a minimum.

Consider the orthogonal coordinates , , of a surface represented as one-valued regular functions[4] of two parameters and . If we consider these as the rectangular coordinates of a point on the plane, then to every point of the surface there will correspond a definite point of the -plane, and these points in their collectivity fill out a definite portion of the plane, which may be looked upon as the image of the surface on the plane. To every curve on the surface corresponds a curve in this part of the -plane and reciprocally.

Further, consider and as one-valued functions of a quantity ; hence, to every value there corresponds a point of the -plane, and therefore, also, in case this point lies in the definite portion of the -plane, there is a corresponding definite point of the surface.

Consequently if and are values of which correspond to the two fixed points on the surface, then the length of any curve which lies between these two points is determined through

,

where

,
,
.

We have then to determine and as functions of , so that is a minimum.

Article 11.
In the case of the above problem it is necessary to apply the representation there given, whereas in Problem I and Problem II the expression of and as one-valued functions of may be regarded as expedient. In Problem III the variables and must he regarded as functions of a third variable. We cannot regard as a function of , for we know nothing about the trace of the curve. If we wished to regard only as a double-valued function of , we would even then encounter many difficulties. Hence the requirements must be made that and be so determined as one-valued functions of , that the integral in the preceding article be a minimum.

Article 12.
Problem IV. Find the form of the surface of rotation, which, having an axis lying in a fixed direction, offers the least resistance in moving through a liquid in the direction of the axis, it being supposed that the resistance of an element of surface is proportional to the square of the component of velocity in the direction of its normal.

This problem is due to Newton.[5]

It is assumed that the friction between the body and the fluid and that within the fluid itself may be neglected.

Let the -axis be the axis of rotation, an element of the generating curve, the angle between the normal and the -axis, so that .

A zone of the surface is therefore given by

.

The component of velocity in the normal direction is , and the resistance in the normal direction which the zone offers, is

.

This quantity multiplied by gives the resistance in the direction of the -axis. We consequently have the required resistance of the body expressed by the integral

.

Our problem then is to connect two points and by a curve so that the zone which it generates about the -axis offers the least resistance. Neglecting the constant factor 2\pi v^{2}, we have to determine and as one-valued functions of so that the integral

shall be a minimum.

Article 13.
That which is common to the four problems stated above consists in the determination of and as one-valued functions of a quantity in such a way that an integral dependent upon them of the form

will have the smallest possible value. Here and have fixed values so that the corresponding coordinates , of the initial and the final point of the curve are supposed to be known.

represents a one-valued regular function of the four arguments , , , of which and (since they represent the direction of the tangent to the curve) are to be regarded as unrestricted, while the region of the point , may be either the whole plane or only a continuous portion of it.

Article 14.
The condition that , should have fixed values is not essential; moreover both end-points may move, as in the case of the third problem, if we give it the following form : Two curves are given on a surface; among all the possible curves between the points of the one curve and the points of the other, that curve is to be found which has the shortest length. We are accustomed to call this the geodesic distance of two curves.

In order to solve this problem, we must first solve the special Problem III, since, if a curve has the property of being of minimum length such as is required above, it must also retain the same property, if we consider the end-points fixed. Hence from III the nature of the curve must be determined. The variation of the endpoints gives in addition certain special properties which the curve must possess.

For example, the shortest distance between two curves which lie in the same plane is clearly a straight line; through the variation of the end-points it follows that this straight line must be perpendicular to both curves at the same time.

Article 15.
Essentially different from the four problems already given is the following:

Problem V. It is required to draw a closed curve which with a given periphery inscribes the greatest possible area.

Let and be one-valued functions of , say and , such that for two definite values and of the corresponding points , of the curve coincide, and that, if goes from a smaller value to a greater value , the point , completely traverses the curve in the positive direction. Then twice the area of the surface included by the curve is expressed by the integral

,

and the periphery of the curve is given by the integral

.

Our problem then is: So determine and as one-valued functions of that shall have the greatest possible value, while at the same time has a given value.

Article 16.
Problem VI. What form is taken by an indefinitely thin, absolutely flexible, but inexpansible thread which is fixed at both ends, if the action of gravity alone acts upon it?

This problem offers the characteristics of a minimum, for with stable equilibrium the center of gravity must be as low as possible. If the -axis is taken vertical with the direction upward, and if denotes the length of the curve, and , the coordinates of the center of gravity, then is determined from the equation

,

where

.

The problem may be stated thus: the variables and are to be determined as one-valued functions of a quantity in such a way that the first of the above integrals has a minimum value, while the second retains a given fixed value.

Article 17.
Problems V and VI are usually classified under the name, Relative Maxima and Minima, a term which requires no further explanation. In general they are included in the following problem: Let and be two functions of the same character as the function of Art. 13. It is required to determine and as one valued functions of a quantity in such a way that the integral

has a maximum or a minimum value, while at the same time the integral

conserves a given value.

Article 18.
We shall give in the sequel what we believe to be a rigorous treatment of the problems already formulated. The reader may propose for himself natural extensions of what is given; for example, instead of taking two variables, consider an integral having as integrand a function of variables. Further, subject these variables to subsidiary conditions and also allow the second and higher derivatives of the variables with respect to a quantity to enter the discussion. Then double integrals which lead to the study of Minimal Surfaces may be treated by methods of variation (see Arts. 175 et seq.).

Article 19.
We may define the object of the Calculus of Variations in a still more general manner by the introduction of a fundamental conception, that of the variation of a curve. In former times the Calculus of Variations was considered one of the most difficult branches of analysis. It was wrongly thought that the difficulty was in the supposed lack of clearness in the fundamental conceptions, especially in that of the variation of a curve. The difficulties that arise are mostly in other directions.

In the Theory of Maxima and Minima we say that for a definite system of values of the variables the value of a function is a maximum or a minimum, if this value of the function for this system of values is greater or smaller than it is for all the neighboring systems of values.

We say[6] of a function of one variable, it has, at a definite position , a maximum or a minimum value, if this value for is respectively greater or less than it is for all other values of which are situated in the neighborhood of as near as we wish to .

The analytical condition that shall have for the position

a maximum, is expressed by for

a minimum, is expressed by for

In the same way we say a function of variables has at a definite position , a maximum or a minimum, if the value of the function for is respectively greater or smaller than it is for all other system,s of values which are situated in the neighborhood as near as we wish to the first position.

As here we speak of a neighboring system of values, so also we speak in the Calculus of Variations of curves which lie in the neighborhood of a given curve; and we require that an integral in the case of a minimum should be less and in the case of a maximum greater when taken over the given curve than for any of the neighboring curves.

In order to fix the conception of a neighboring curve, and to make clear the analogy of the same with the conception of a neighboring system of values, let us consider first, instead of the given curve, a broken line A-, and let us cause the same to slide just a little from its original position.

Then in the new position every corner will correspond to a definite corner in the old position, and moreover the new position will be as little different from the old position as we wish, if we stipulate that the distance between any two corresponding points and shall be smaller than any quantity , where is as small as we choose. Now, by increasing the number of sides, let the broken line pass into the given curve; then the points will also form a curve which is little different from the first curve, and which we consequently call neighboring to the first curve.

Therefore we can say a curve is neighboring to another curve, or exists out of another curve through a variation[7] as small as we choose, if to every point of the latter curve there corresponds a definite point on the former curve, and also the distance between any two corresponding points is smaller than , where is as small as we wish.

The geometrical conception of a neighboring curve offers no obscurity. In a similar manner it is easy to see that for every change of the curve there is a corresponding change of the integral

,

and that this change will be indefinitely small when the second curve is neighboring to the first.

This change of the value of the integral must of course be a continuous, negative one if the integral is to be a maximum, and a continuous, positive one if the integral is to be a minimum.

Article 20.
Observing what has just been said, we may formulate the problems of Arts. 13 and 17 as follows:

The variables and are to be determined as one-valued functions of a quantity in such a way that when we define a curve by the equations , , and cause the curve to vary as little as we wish, the change which thereby takes place in the integral

must be continuously positive if a minimum is to enter, and continuously negative if we require a maximum.

In the case of Relative Maxima and Minima, for every indefinitely small variation of the curve for which the integral

conserves its value unchanged, the integral

according as to whether a maximum or a minimum is to be present, must be constantly smaller or constantly greater than for the curve which is given by the equations ,.

Article 21.
We must seek strenuous methods for the solution of the problems presented above. The methods by means of which Jacobi and the older mathematicians, Bernoulli and his contemporaries, Newton and Leibnitz, sought to solve these questions lead only to the formation of certain differential equations and in propitious cases to the integration of such equations. But these methods were not sufficient for a definitive determination as to whether the curve which had been found in reality offered the required properties.

We know that in the problems of the ordinary Theory of Maxima and Minima it is not always necessary that a maximum or a minimum exist[8] It is certain that every variable has an upper and a lower limit within any region for which this variable has a meaning. Therefore there exists a limit such that all values which a variable can assume are greater than , and that everywhere in the neighborhood of there are values which the variable can assume. We call the lower limit of the variable. In the same way there is an upper limit. These limits need not always be reached. There are consequently two cases possible: Either the values which are denoted as upper and lower limits may in reality be reached by the variables, or the variables may only come indefinitely near without ever reaching these limits. It is therefore inadmissible to presuppose the existence of a maximum or a minimum. For example, Newton's problem, cited above, has no solution, and in the case of the first problem there is sometimes a minimum and sometimes no such minimum exists.

Problems.

It is suggested that the student select two or three of the following problems and apply the same methods of solution to them as will be done for the six problems already proposed.

1. Problem of least action. Find the minimum value of the integral

the limiting values of and . being fixed, and determine under what conditions a parabolic are' is in reality a solution of the problem. The problem may also be stated as follows: Determine the path of a particle for which the action is a minimum between fixed points, if the velocity at any point is that due to a fall from a straight line , the axis of being vertically downwards. [See Todhunter, Researches in the Calculus of Variations, p. 147; see also Quarterly Journal of Mathematics, Nov., 1868.] The discontinuity here is very similar to that which we shall find in the case of Problem I, p. 1.

2. Principle of least action in the elliptic motion of a planet. A particle is projected from a given point with a given velocity and is attracted to a fixed point by a force varying inversely as the square of the distance. Determine the path of minimum action to a second fixed point. [See Todhunter, Researches, etc., p. 160; Todhunter, History of the Calculus of Variations, p. 251; Jellett, Calculus of Variations, p. 76; Jacobi, Crelle, vol. 17, p. 68; Liouville's Journ., tom. III, p. 44; Delaunay, Liouville's Journ., tom. VI, p. 209.]

3. Determine the curve which renders the integral a maximum, when the variables are given fixed limits. [See Euler, Methodus Inveniendi Lineas Curvas Maximi Minimive Gaudentes, Lausanne, 1794, p. 52; Woodhouse, A Treatise on Isoperimetrical Problems and the Calculus of Variations, p. 124.]

4. Given the length of a curve, determine its nature, when the volume generated by its rotation about a fixed axis is a maximum or a minimum. [See Euler, Methodus, etc., p. 196; Woodhouse, A Treatise, etc., p. 125; Moig-no et Lindelof, Calcul de Variations, p. 216; Jellett, Calculus of Variations, p. 160.]

5. Required the curve that, by a revolution about a fixed axis, generates the greatest or the least volume, the surface area being constant. [See Euler, Methodus, etc., p. 194; Moigno et Lindelof, Calcul de Variations, p. 218; Delaunay, Liouville's Journ., torn. VI, p. 315; Phil. Mag., 1866; Todhunter, Researches, etc., p. 68; Jellett, Calculus of Variations, p. 161 and note, p. 364.]

6. Find the curve which generates by its rotation the solid of greatest volume, the length of the curve and its area being given. [See Lacroix, Calc. Diff'l et Int., Vol. II, p. 713; consult further the references above for this and the following problems.]

7. Find the curve of quickest descent when the length of the curve is given. [See John Bernoulli's Works, Vol. II, p. 255; Memoires de I'Acadmie des Sciences, Paris, 1718, p. 120.]

8. A plane curve being given, determine a second curve of given length such that the area inclosed between the two curves be a maximum.

9. Among all curves of the same length, find the one which, by its revolution about an axis, will generate the greatest or the smallest surface area.

10. It is required to maximize or minimize the integral

,

where is a given function of the variables , , , which are connected by the equation , being a known function.

11. Find the curve of minimum length between two fixed points in space, the radius of curvature being a constant.

12. Find the form which a homogeneous body of given volume must take that its attraction upon a material point in a definite direction be as great as possible.

  1. Poincaré (Bulletin de la Société Mathématique de France, T XI. 1883.) See also my lectures on the Theory of Maxima and Minima, etc. Page 13.
  2. Woodhouse (A Treatise on Isoperimetrical Problems and the Calculus of Variations, 1810) writes (p. 1): "The ordinary questions of maxima and minima were amongst the first that engaged the attention of mathematicians at the time of the invention of the Differential Calculus (1684), three years before the publication of the Principia. The first problem relative to a species of maxima and minima distinct from the ordinary was proposed by Newton in the Principia; it was that of the solid of least resistance. But the subject became not matter of discussion and controversy till John Bernoulli (Acta Erudit., 1696, p. 269) required the curve of quickest descent."
  3. This is, of course, different from the time of the preceding article.
  4. See my lectures on the Theory of Maixma and Minima, etc. Page 31.
  5. Newton, Principia, Book II, prop. 34. Thus Newton was the first to consider a problem in the Calculus of Variations, and his problem involved a discontinuous solution. Solutions of it have been given by Euler and almost all other writers on the Calculus of Variations. We shall see that one of the principal conditions for a minimum (the condition of Weierstrass) is not satisfied, and that there can never be a maximum or a minimum.
  6. See Lectures on the Theory of Maxima and Minima of Functions of Several Variables, p. 32.
  7. The notion of the variation of a curve was first introduced by Lagrange. He considered the required curve transposed into one that lies indefinitely near it by writing instead of each point , of the curve another point , . This operation of transition he called a variation.
  8. See Lectures, etc. (loc. cit.), p. 86.