Calculus of Variations/CHAPTER IV

CHAPTER IV: PROPERTIES OF THE FUNCTION $F(x,y,x',y')$.

• 62 The function $F$ defined as a function of its arguments.
• 63,64,65,66,67 Necessary conditions and sufficient conditions.
• 68 The function $F(x,y,x',y')$ must be homogeneous of the first degree in $x'$ and $y'$.
• 69 Integrability of the function $F(x,y,x',y')$.
• 70 The integral $I=\int _{x_{0}}^{x_{1}}F\left(x,y,1,{\frac {{\text{d}}y}{{\text{d}}x}}\right){\text{ d}}x$, when $x$ and $y$ are one-valued functions of each other.
• 71 Introduction of the variable $t$ or $s$.
• 72 Analytical condition for the function $F$.
• 73 Introduction of the function $F_{1}$.

Article 62.
Consider the general integral of Art. 13:

$I=\int _{t_{0}}^{t_{1}}F(x,y,x',y')~{\text{d}}t$,

where $F$ is a given function of the four arguments, $x$, $y$, $x'$, $y'$, the quantities $x'$ and $y'$ being written for ${\text{d}}x/{\text{d}}t$ and ${\text{d}}y/{\text{d}}t$; further we must regard $F$ as a one-valued regular function of these four arguments, one-valued not in the analytical sense, but only for real values of the arguments; $x$ and $y$ are defined for the whole plane or for a connected portion of it, while $x'$ and $y'$ are to be considered as variables that are not limited, since they determine the direction of the tangent, and it is supposed that we may go in any direction from the point $x$, $y$. In our problem new assumptions are made regarding $x$ and $y$, but not regarding $x'$ and $y'$ We further assume that the functions $x$, $y$, $x'$ and $y'$, are capable of being differentiated, and that the curve is regular throughout its whole extent, or is composed of regular portions. Consequently $x$ and $y$ considered as functions of $t$ and written $x(t)$, $y(t)$ are one-valued regular functions of $t$ throughout its whole extent or throughout the regular portions; in the latter case we shall limit ourselves to one regular portion. If we did not make this assumption, the curve could not be the subject of mathematical investigation, since there is no method of treating irregular curves in their generality; and, if we wish the rules of the differential and integral calculus to be sufficient, then we must first apply our investigation to such functions, to which the rules are applicable without any limitation; that is, to functions having the above properties.

Article 63.
If we find a curve which is regular and which satisfies the conditions of the problem, then it still remains as a supplement to prove that it is the only curve which satisfies the conditions of the problem.

For example, it is found that of all regular closed curves of given perimeter the circle is the one which encloses the greatest surface area; a priori, however, it is not known that a regular curve satisfies the problem. We know that of all polygons with a given number of sides and having a given perimeter the regular polygon has the greatest surface area, and we thus come to the conclusion that the circle, to which the polygon approaches when the number of sides is increased, will have the greatest surface area of all the closed curves ; however, no one will recognize in this a rigorous proof, and in fact there still remains a peculiar artifice to prove this property of the circle.

Article 64.
The chief difficulty in all analysis consists in giving a strenuous proof that the necessary conditions, that have been found for the existence of a certain property, are also sufficient. In analytical researches we make conclusions in the following manner : If the analytical quantities exist, which are required through the problems that have been set, then they must have certain properties ; this gives the necessary conditions for the sought functions. It remains yet reciprocally to prove : If the conditions for an analytical object (curve, surface, etc. ) are fulfilled, then the analytical object satisfies the conditions of the problem.

We therefore presuppose in our investigations, that the required functions are regular in their whole extent, and we seek the necessary conditions for the function which are given from the problems. Finally we will free ourselves from the limitations as far as it is possible, and see whether also the functions which have been found correspond to the conditions of the problem.

Article 65.
The development of a mathematical idea is, as a rule, first suggested by a concrete instance. We assume, for example, the existence in nature of something which we call the area of a limited plane. This area we express by a mathematical formula. We extend our formula and talk of the area of a curved surface. The mathematical formula exists. That to which it corresponds in nature may or may not have an objective existence. The word "area," however, is defined for us, and is limited by the mathematical formula. When the formula ceases to be intelligible, ceases to have a meaning and to give a value, then also does the idea "area" cease to exist for us. We must always presuppose those limitations to be involved in our symbols which permit of the formula having a meaning.

Article 66.
Only for regular curves do we compare our integrals; for such curves alone have they a meaning. Among this class of curves we seek one which gives a maximum or a minimum value of our integral. And when we put our theory into practice we assume the non-existence of quantites other than those which our theory has actually compared. Here we run a risk.

It may be that in some particular problem we have assigned a certain role; it may be also that, as far as our theory goes, we are correct in assuming the possibility of the existence of all the regular curves that are compared with one another and that their roles relative to one another has not been misstated. But it may be that there exists in nature the possibility of quantities other than those defined by our definite integrals along regular curves, and these quantities may have the same essential properties relative to the problem in question as our various definite integrals. It may be also that to one of these quantities nature has assigned that very role which we have been seeking among our definite integrals.

When we apply any mathematical theory to objective reality, we make assumptions in the way of continuity, differentiation, etc., regarding the possibilities which are permitted in nature. The question arises, do our hypotheses include all possibilities?

Article 67.
We may emphasize the fact that in the development of a general theory, as a rule its scope is not determined beforehand. The quantities and functions to which we must apply the operations involved are named a priori, but formulas are developed on the supposition that the operations involved are feasible and have a meaning. The scope of the formulae is afterwards defined by the territory in which all the steps involved have some signification, or by the exclusion of any realm in which they would be incapable of interpretation.

Article 68.
We will now prove some important properties of the function $F$ (Art. 62). In the problems which we have discussed the following is to be observed : the value of the integral, which is to be a minimum, depends in all cases only upon the form of the curve which is to be determined, not upon the manner in which $x$, $y$ are represented as functions of a quantity $t$.

For example, if in the first problem we write the integral in the form

$\int _{x_{0}}^{x_{1}}y{\sqrt {1+\left({\frac {{\text{d}}y}{{\text{d}}x}}\right)^{2}}}~{\text{d}}x$,

then $t$ is exactly equal to $x$, and it is clear that the value of this integral is the same as it was for the previous form (Art. 7).

If we write for $t$ any function of another quantity $\tau$ of such a nature that to the values $t_{0}$ and $t_{1}$ of $t$ the values $\tau _{0}$ and $\tau _{1}$ of $\tau$ correspond, and that the curve with increasing $\tau$ will be traversed in the same direction as in the first case with increasing $t$, then the integral must remain unaltered, if it is to be independent of the manner in which $x$, $y$ are represented as functions of the quantity $t$; that is, we must have as the integral $I$ of Art. 62

$1)\qquad \int _{t_{0}}^{t_{1}}F\left(x,y,{\frac {{\text{d}}x}{{\text{d}}t}},{\frac {{\text{d}}y}{{\text{d}}t}}\right)~{\text{d}}t=\int _{\tau _{0}}^{\tau _{1}}F\left(x,y,{\frac {{\text{d}}x}{{\text{d}}\tau }},{\frac {{\text{d}}y}{{\text{d}}\tau }}\right)~{\text{d}}\tau$.

The simplest function of this kind that we can write for $t$, is $t=k\tau$, where $k$ represents any arbitrary but positive quantity. Hence considering $x$, $t$ as functions of $\tau$ in the left-hand side of 1), we have

$\int _{t_{0}}^{t_{1}}F\left(x,y,{\frac {{\text{d}}x}{{\text{d}}t}},{\frac {{\text{d}}y}{{\text{d}}t}}\right)~{\text{d}}t=\int _{\tau _{0}}^{\tau _{1}}F\left(x,y,{\frac {1}{k}}{\frac {{\text{d}}x}{{\text{d}}\tau }},{\frac {1}{k}}{\frac {{\text{d}}y}{{\text{d}}\tau }}\right)~k{\text{d}}\tau$;

hence

$:<math>2)\qquad \int _{\tau _{0}}^{\tau _{1}}F\left(x,y,{\frac {{\text{d}}x}{{\text{d}}\tau }},{\frac {{\text{d}}y}{{\text{d}}\tau }}\right)~{\text{d}}\tau =\int _{\tau _{0}}^{\tau _{1}}F\left(x,y,{\frac {1}{k}}{\frac {{\text{d}}x}{{\text{d}}\tau }},{\frac {1}{k}}{\frac {{\text{d}}y}{{\text{d}}\tau }}\right)~k{\text{d}}\tau$.

Since this equation must be true for any arbitrary positive vakie of $k$, which however is not necessarily a constant, but may be any continuous positive function, it follows that the functions to be integrated must themselves be equal for every positive value of $k$; and consequently

$<F\left(x,y,{\frac {{\text{d}}x}{{\text{d}}\tau }},{\frac {{\text{d}}y}{{\text{d}}\tau }}\right)=kF\left(x,y,{\frac {1}{k}}{\frac {{\text{d}}x}{{\text{d}}\tau }},{\frac {1}{k}}{\frac {{\text{d}}y}{{\text{d}}\tau }}\right)$

or, writing $k=1/k$.

$3)\qquad F(x,y,kx',ky')=kF(x,y,x',y')$.

That is, if the integral $I$ is to depend only upon the form of the curve (or in other words, upon the analytical connection between $x$ and $y$), then $F(x,y,x',y')$, with regard to $x'$ and $y'$ must be a homogeneous function of the first degree. This condition is also sufficient to assure that the integral depends only upon the form of the curve; for consider $x$,$y$ first expressed as functions of a quantity $t$ and then as functions of a quantity $\tau$, and if these functions are of such a nature that the curve is traversed from the beginning-point to the end-point when $t$ takes all values from $t_{0}$ to $t_{1}$, and $\tau$ all values $\tau _{0}$ to $\tau _{1}$, then we can write ${\text{d}}t/{\text{d}}\tau =k$, if $t$ increases at the same time as $\tau$. Since $k$ is a positive quantity, the correctness of the expression 2) follows from the existence of 3) and at the same time also the correctness of 1).

It follows also that

${\frac {\partial F(x,y,kx',ky')}{\partial (kx')}}=k{\frac {\partial F(x,y,x',y')}{\partial (kx')}}={\frac {\partial F(x,y,x',y')}{\partial x'}}=F^{(1)}(x,y,x',y')$, say.

In the same way the partial derivative of $F$ with respect to its fourth argument is invariantive and may be denoted by $F^{(2)}(x,y,x',y')$.

Article 69.
The condition that $F(x,y,x',y')$ must be a homogeneous function of the first degree with regard to $x'$ and $y'$ is generally expressed in another manner. In fact, it is nothing else than the condition of integrability of $F$. For if $F(x,y,x',y'){\text{d}}t$ is to be an exact differential, so that, say, $F(x,y,x',y')={\text{d}}\phi$, then the equation

$F(x,y,x',y')=(\partial \phi /\partial x)x'+(\partial \phi /\partial y)y'+(\partial \phi /\partial x')x''+(\partial \phi /\partial y')y''$

must exist identically.

Since no second differential quotient is present in $F$, it follows that $\partial \phi /\partial x'=0$ and $\partial \phi /\partial y'=0$, i.e., $\phi$ does not contain explicitly $x'$ or $y',$ and therefore

$F(x,y,x',y')=(\partial \phi /\partial x)x'+(\partial \phi /\partial y)y'$

But this is nothing more than that $F$ is a homogeneous function of the first degree in $x'$, $y'$.

This is everywhere the case in the examples given in Chap. I.

Article 70.
If the curve is of such a nature that one may regard the one coordinate as a one-valued function of the other and in such a way that for every value of $x$ between two limits $x_{0}$ and $x_{1}$, there corresponds only one definite value of $y$, and that $x$ continuously increases when we traverse the curve from the beginning-point to the end-point, then we may choose for $t$ the quantity $x$ itself, and therefore write the integral in the form

$4)\qquad I=\int _{x_{0}}^{x_{1}}F(x,y,1,{\text{d}}y}{{\text{d}}x)~{\text{d}}x$,

as it is usually written.

Article 71.
This representation is not always true, since the above conditions which are necessary are not always fulfilled; for example, in the fourth problem of Chapter I we must distribute the )-et unknown curve into several parts, and this is not always convenient.

On the other hand, a representation such as given above is always possible, if we introduce the quantity $t$, since one could introduce as the variable $t$ the arc $s$ of the curve measured from the beginning-point. Besides in the form 4) it sometimes unavoidably happens that ${\text{d}}y/{\text{d}}x$ and consequently $F$ becomes infinite within the limits of integration; on the other hand it is generally possible so to choose $t$ that this is not the case.

For these reasons, in spite of the fact that many developments become more cumbrous, it is preferable to treat the integral I in the form

$I=\int _{t_{0}}^{t_{1}}F(x,y,x',y')~{\text{d}}t$;

for on the other hand its great symmetry overbalances the fault just mentioned.

Article 72.
Analytical condition for' $F(x,y,x',y')$.

In the relation (Art. 68)

$F(x,y,kx',ky')=kF(x,y,x',y')$,

write $k=1+h$, then is

$F[x.y.(1+h)x',(1+h)y']=(1+h)F(x,y,x',y')$,

or,

$F+\left({\frac {\partial F}{\partial x'}}x'+{\frac {\partial F}{\partial y'}}y'\right)h+h^{2}(\cdots )=F+hF$,

where

$F\equiv F(x,y,x',y')$.

Therefore equating the coefficients of $h$:

$F=x'{\frac {\partial F}{\partial x'}}+y'{\frac {\partial F}{\partial y'}}{\text{,}}\qquad {\text{[1]}}$

which again is the condition of homogeneity.

Article 73.
DifEerentiate the above equation [1] first with regard to $x'$ and then with regard to $y'$, which is allowable, since $F$ is a regular function, and $x'$, $y'$ vary in a continuous manner, and we have

$\alpha )\qquad {\frac {\partial ^{2}F}{\partial x'^{2}}}x'+{\frac {\partial ^{2}F}{\partial y'\partial x'}}y'=0$,

$\qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad {\text{[2]}}$

$\beta )\qquad {\frac {\partial ^{2}F}{\partial x'\partial y'}}x'+{\frac {\partial ^{2}F}{\partial y'^{2}}}y'=0$.

Hence, from $\alpha )$

${\frac {\partial ^{2}F}{\partial x'^{2}}}:{\frac {\partial ^{2}F}{\partial y'\partial x'}}=y':-x'=y'^{2}:-x'y'$,

and from $\beta )$

${\frac {\partial ^{2}F}{\partial x'\partial y'}}:{\frac {\partial ^{2}F}{\partial y'^{2}}}=-y':x'=-x'y':-x'^{2}$:

therefore

${\frac {\partial ^{2}F}{\partial x'^{2}}}:{\frac {\partial ^{2}F}{\partial x'\partial y'}}:{\frac {\partial ^{2}F}{\partial y'^{2}}}=y'^{2}:-x'y':x'^{2}$;

and, if $F$ denotes the factor of proportionality, we have:

${\frac {\partial ^{2}F}{\partial x'^{2}}}=F_{1}y'^{2}$; ${\frac {\partial ^{2}F}{\partial x'\partial y'}}=-F_{1}x'y'$; ${\frac {\partial ^{2}F}{\partial y'^{2}}}=F_{1}x'^{2}{\text{,}}\qquad {\text{[3]}}$

and consequently

${\frac {\frac {\partial ^{2}F}{\partial x'^{2}}}{y'^{2}}}={\frac {\frac {\partial ^{2}F}{\partial x'\partial y'}}{-x'y'}}={\frac {\frac {\partial ^{2}F}{\partial y'^{2}}}{x'^{2}}}=F_{1}$.

$F$ is of the first dimension in $x'$ and $y'$; ${\frac {\partial F}{\partial x'}}$,${\frac {\partial F}{\partial y'}}$ are of the dimension 0 in $x'$ and $y'$; ${\frac {\partial ^{2}F}{\partial y'^{2}}}$, ${\frac {\partial ^{2}F}{\partial x'y'}}$, ${\frac {\partial ^{2}F}{\partial x'^{2}}}$ are of the -1st dimension in $x'$ and $y'$; consequently $F_{1}$ is of the dimension -3 in $x'$ and $y'$.

This function $F_{1}$ plays an exceedingly important role in the whole theory.