Calculus of Variations/CHAPTER II

CHAPTER II: EXAMPLES OF SPECIAL VARIATIONS OF CURVES. APPLICATIONS TO THE CATENARY.

• 22 Total variation in the case of Problem I, Chapter I.
• 23 A bundle of neighboring curves.
• 24 The first variation.
• 25 The integral $I=\int _{x_{0}}^{x_{1}}F\left(x,y,{\frac {{\text{d}}y}{{\text{d}}x}}\right){\text{ d}}x$
• 26 The vanishing of the first variation.
• 27 Application to Problem I.
• 28 The differential equation of this problem.
• 29 The integral $I=\int _{x_{0}}^{x_{1}}F(y,y'){\text{ d}}x$
• 30 Solution of the differential equation of Art. 26.
• 31 The notion of a region within which two neighboring curves do not intersect.
• 32 The catenary.

Article 22.
Let us consider again the integral of Art. 6,

${\frac {S}{2\pi }}=\int _{x_{0}}^{x_{1}}y{\sqrt {1+\left({\frac {{\text{d}}y}{{\text{d}}x}}\right)^{2}}}~{\text{d}}x\qquad {\text{[1]}}$.

Suppose that there is a minimum surface area that is generated by the rotation of a curve between the two fixed points $P_{0}$ and $P_{1}$ and let this curve be $y=f(x)$. Let $\eta$ be the distance between this curve and any neighboring curve measured on the $y$-ordinate, and suppose that $\eta$ is a continuous function of $x$ subject to the conditions: that for $x=x_{0}$, $\eta =0$; for $x=x_{1}$, $\eta =0$; and for all other points $|\eta |<\rho$, where $\rho$ may be as small as we choose.

$\eta '={\frac {{\text{d}}\eta }{{\text{d}}x}}$ and $\int _{x_{0}}^{x_{1}}\eta '~{\text{d}}x=[\eta ]_{x_{0}}^{x_{1}}=0$

The integral of any neighboring curve corresponding to [1] is

$\int _{x_{0}}^{x_{1}}(y+\eta ){\sqrt {1+\left({\frac {{\text{d}}(y+\eta )}{{\text{d}}x}}\right)^{2}}}~{\text{d}}x\qquad {\text{[2]}}$.

Hence the total variation caused in [1] when, instead $y=f(x)$, we take a neighboring curve, is

${\frac {\Delta S}{2\pi }}=\int _{x_{0}}^{x_{1}}(y+\eta ){\sqrt {1+\left({\frac {{\text{d}}(y+\eta )}{{\text{d}}x}}\right)^{2}}}~{\text{d}}x-\int _{x_{0}}^{x_{1}}y{\sqrt {1+\left({\frac {{\text{d}}y}{{\text{d}}x}}\right)^{2}}}~{\text{d}}x{\text{.}}\qquad {\text{[3]}}$

$\Delta S$ has always a positive sign, since the surface in question is a minimum.

Article 23.
Instead of the one neighboring curve, we may consider a whole bundle of such curves, if for $\eta$ we substitute $\epsilon \eta$, where $\epsilon$ is independent of $x$ and has any value between $-1$ and $+1$. The expression [3] becomes then

${\frac {\Delta S}{2\pi }}=\int _{x_{0}}^{x_{1}}(y+\epsilon \eta ){\sqrt {1+\left({\frac {\text{d}}{{\text{d}}x}}(y+\epsilon \eta )\right)^{2}}}~{\text{d}}x-\int _{x_{0}}^{x_{1}}y{\sqrt {1+\left({\frac {{\text{d}}y}{{\text{d}}x}}\right)^{2}}}~{\text{d}}x{\text{;}}\qquad {\text{[4]}}$

and, developing $\Delta S$ by Taylor's Theorem,

$\Delta S=\epsilon \delta S+{\frac {\epsilon ^{2}}{2!}}\delta ^{2}S+{\frac {\epsilon ^{3}}{3!}}\delta ^{3}S+\cdots \qquad {\text{[5]}}$

There is no constant term in this last development, since when $\epsilon$ is made zero in [4] the first and second integrals cancel each other.

$\delta S$ is known as the first variation,

$\delta ^{2}S$ is called the second variation, etc.

Instead of taking $\eta$ a very small quantity, we may take $\epsilon$ so small that $\epsilon \eta$ is as small as we choose.

With Lagrange (Misc. Taur., tom. II, p. 174), writing $\eta =\delta y$, it is seen that the total change in $y$ is $\epsilon \eta =\epsilon \delta y=\Delta y$.

REMARK. The sign of differentiation and the sign of variation may be interchanged; for example, the 1st derivative of a variation is equal to the 1st variation of a derivative, as is seen by writing

$\eta =\delta y$, then $\eta '=(\delta y)'={\frac {\text{d}}{{\text{d}}x}})(\delta y).\qquad {\text{[i]}}$

Again $\eta =\delta y$; change $y$ into $y+\epsilon \eta$, and consequently $y'$ into $y'+\epsilon \eta '$. Hence $\eta '$ is the first variation of $y'$, so that

$\eta '=\delta y'=\delta \left({\frac {{\text{d}}y}{{\text{d}}x}}\right){\text{;}}\qquad {\text{[ii]}}$

and therefore from [i] and [ii]

${\frac {\text{d}}{{\text{d}}x}}(\delta y)=\delta \left({\frac {{\text{d}}y}{{\text{d}}x}}\right)$.

It follows too that owing to the presupposed existence of $\eta '$, we must also assume the existence of the second differential coefficient of $y$.

Article 24.
Returning to [4], write $y'={\frac {{\text{d}}y}{{\text{d}}x}}$, $\eta '={\frac {{\text{d}}\eta }{{\text{d}}x}}$. Then expanding the expression under the sign of integration

$(y+\epsilon \eta ){\sqrt {1+(y'+\epsilon \eta ')^{2}}}-y{\sqrt {1+y'^{2}}}$,

we have

$\epsilon \left[\eta {\sqrt {1+(y'+\epsilon \eta ')^{2}}}+{\frac {(y+\epsilon \eta )(y'+\epsilon \eta ')\eta '}{\sqrt {1+(y'+\epsilon \eta ')^{2}}}}\right]_{\epsilon =0}+\epsilon ^{2}(\ldots )$.

Hence, equating the coefficients of the 1st power of $\epsilon$ in [4] and in [5] we have

${\frac {\delta S}{2\pi }}=\int _{x_{0}}^{x_{1}}\left({\sqrt {1+y'^{2}}}\eta +{\frac {yy'}{\sqrt {1+y'^{2}}}}\eta '\right)~{\text{d}}x$,

which is a homogeneous function of the first degree in $\eta$ and $\eta '$. The quantity $\eta '$ cannot be indefinitely large, since then the development would not be necessarily convergent; but see Art. 116.

In a similar manner we may find a definite integral for the second variation, in which the integrand is an integral homogeneous function of the second degree in $\eta$ and $\eta '$; similarly for the third variation, etc.

Article 25.
As a form of the integrals which were given in Problems I, II, III and IV of the preceding Chapter, consider the integral

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

where $F(x,y,y')$ is a known function of $x$, $y$ and $y'$, and where the limits of this integral, $x_{1}$ and $x_{0}$, are fixed. Hence, as above,

$\Delta I=\int _{x_{0}}^{x_{1}}F(x,y_{\epsilon }\eta ,y'+\epsilon \eta ')~{\text{d}}x-\int _{x_{0}}^{x_{1}}F(x,y,y')~{\text{d}}x=\int _{x_{0}}^{x_{1}}[F(x,y_{\epsilon }\eta ,y'+\epsilon \eta ')-F(x,y,y')]~{\text{d}}x$.

This expression, when expanded by Taylor's Theorem, is

$\Delta I=\int _{x_{0}}^{x_{1}}\left({\frac {\partial F}{\partial y}}\epsilon \eta +{\frac {\partial F}{\partial y'}}\epsilon \eta '+\epsilon ^{2}(\cdots )+\cdots \right)~{\text{d}}x$.

We also have, as in Art. 23,

$\Delta I=\epsilon \delta I+{\frac {\epsilon ^{2}}{2!}}\delta ^{2}I+\cdots$;

and by comparing the coefficients of $\epsilon$ in these two expressions, it follows that

$\delta I=\int _{x_{0}}^{x_{1}}\left({\frac {\partial F}{\partial y}}\eta +{\frac {\partial F}{\partial y'}}\eta '\right)~{\text{d}}x\qquad {\text{(A)}}$

In the particular case given in Art. 22, $F=y{\sqrt {1+y'^{2}}}$. Hence

${\frac {\partial F}{\partial y}}={\sqrt {1+y'^{2}}}$ and ${\frac {\partial F}{\partial y'}}={\frac {yy'}{\sqrt {1+y'^{2}}}}$;

and when these relations are substituted in (A) we have, as in Art. 24,

$\Delta I=\int _{x_{0}}^{x_{1}}\left({\sqrt {1+y'^{2}}}\eta +{\frac {yy'}{\sqrt {1+y'^{2}}}}\eta '\right)~{\text{d}}x$

Article 26.
From the relation

$\Delta I=\epsilon \delta I+{\frac {\epsilon ^{2}}{2!}}\delta ^{2}I+\cdots$

it is seen that when $\epsilon$ is taken very small, $\epsilon ^{2}$ is as near as we wish to zero; and consequently when $\epsilon$ is positive and indefinitely small, $\Delta I$ is positive. On the other hand, when $\epsilon$ is indefinitely small and negative, $\Delta I$ is negative.

Hence the total variation $\Delta I$ of the integral will be either positive or negative according as $\epsilon$ is positive or negative, so long as $\delta I$ is different from zero; and consequently there can be neither a maximum nor a minimum value of the integral.

We know, however, if $I$ is a maximum $\Delta I$ is always negative, and if $I$ is a minimum $\Delta I$ is always positive; and consequently in order to have a maximum or a minimum value of the integral, $\delta I$ must be zero.

Article 27.
Applying the above result to the example given in Art. 22 we have

$0=\int _{x_{0}}^{x_{1}}\left({\sqrt {1+y'^{2}}}\eta +{\frac {yy'}{\sqrt {1+y'^{2}}}}{\frac {{\text{d}}\eta }{{\text{d}}x}}\right)~{\text{d}}x\qquad {\text{[6]}}$

Integrating by parts,

$\int _{x_{0}}^{x_{1}}{\frac {yy'}{\sqrt {1+y'^{2}}}}~{\text{d}}\eta =\left[{\frac {yy'}{\sqrt {1+y'^{2}}}}\eta \right]_{x_{0}}^{x_{1}}-\int _{x_{0}}^{x_{1}}{\frac {\text{d}}{{\text{d}}x}}\left({\frac {yy'}{\sqrt {1+y'^{2}}}}\right)\eta ~{\text{d}}x$;

and since, by hypothesis (see Art. 22), $\eta =0$ at both of the fixed points $P_{0}$ and $P_{1}$, we have

$\left[{\frac {yy'}{\sqrt {1+y'^{2}}}}\eta \right]_{x_{0}}^{x_{1}}=0$.

Hence [6] may be written

$0=\int _{x_{0}}^{x_{1}}\left({\sqrt {1+y'^{2}}}-{\frac {\text{d}}{{\text{d}}x}}\left({\frac {yy'}{\sqrt {1+y'^{2}}}}\right)\right)\eta ~{\text{d}}x\qquad {\text{[7]}}$

Article 28.
We assert that in the expression above

${\sqrt {1+y'^{2}}}-{\frac {\text{d}}{{\text{d}}x}}\left({\frac {yy'}{\sqrt {1+y'^{2}}}}\right)$

must always be zero between the limits $x_{0}$ and $x_{1}$. For, assuming that the contrary is the case; then, since $\eta$ is arbitrary, we may, with Heine, write

$\eta =(x-x_{0})(x_{1}-x)\left({\sqrt {1+y'^{2}}}-{\frac {\text{d}}{{\text{d}}x}}\left({\frac {yy'}{\sqrt {1+y'^{2}}}}\right)\right)$,

where $\eta$ becomes zero for the valued $x=x_{0}$ and $x=x_{1}$. Substituting this value of $\eta$ in [7], we have

$=\int _{x_{0}}^{x_{1}}\left({\sqrt {1+y'^{2}}}-{\frac {\text{d}}{{\text{d}}x}}\left({\frac {yy'}{\sqrt {1+y'^{2}}}}\right)\right)^{2}(x-x_{0})(x_{1}-x)~{\text{d}}x{\text{,}}\qquad {\text{[8]}}$

an expression which is positive within the whole interval $x_{0}\ldots x_{1}$.

The integrand in [8] , looked upon as a sum of infinitely small elements, has all its elements of the same sign and positive; so that the only possible way for the right-hand member of [8] to be zero is that

${\sqrt {1+y'^{2}}}-{\frac {\text{d}}{{\text{d}}x}}\left({\frac {yy'}{\sqrt {1+y'^{2}}}}\right)=0$

We therefore have a differential equation of the second order for the determination of the unknown quantity $y$.

Article 29.
This differential equation is a special case of the more general differential equation, which may be derived from the integral

$I=F(y,y')~{\text{d}}x$;

whence, as before (Arts. 25 and 27),

$\delta I=\int _{x_{0}}^{x_{1}}\left({\frac {\partial F}{\partial y}}\eta +{\frac {\partial F}{\partial y'}}\eta '\right)~{\text{d}}x=\int _{x_{0}}^{x_{1}}\left({\frac {\partial F}{\partial y}}-{\frac {\text{d}}{{\text{d}}x}}\left({\frac {\partial F}{\partial y'}}\right)\right)\eta ~{\text{d}}x$.

As in Art. 27, we have

${\frac {\partial F}{\partial y}}-{\frac {\text{d}}{{\text{d}}x}}\left({\frac {\partial F}{\partial y'}}\right)=0$

or

${\frac {\partial F}{\partial y}}={\frac {\text{d}}{{\text{d}}x}}\left({\frac {\partial F}{\partial y'}}\right)\qquad {\text{[9]}}$

But

${\text{d}}F(y,y')={\frac {\partial F}{\partial y}}{\text{d}}y+{\frac {\partial F}{\partial y'}}{\text{d}}y'\qquad {\text{[10]}}$

or

${\text{d}}F(y,y')-\left({\frac {\partial F}{\partial y}}{\text{d}}y+{\frac {\partial F}{\partial y'}}{\text{d}}y'\right)=0$

Hence from [9],

${\text{d}}F(y,y')-\left({\frac {\text{d}}{{\text{d}}x}}\left({\frac {\partial F}{\partial y}}\right){\text{d}}y+{\frac {\partial F}{\partial y'}}{\text{d}}y'\right)=0$,

or

${\text{d}}F(y,y')-{\text{d}}\left(y'{\frac {\partial F}{\partial y'}}\right)=0$,

and integrating,

$F(y,y')-y'{\frac {\partial F}{\partial y'}}=C{\text{,}}\qquad {\text{[11]}}$,

where $C$ is the constant of integration.

The relation [11] exists only when the integrand of the given integral does not contain explicitly the variable $x$; otherwise the relation [10] would not be true, and then we could not deduce [11].

Article 30.
Applying this relation [11] to the special case above (Art. 28) where

$F(y,y')=y{\sqrt {1+y'^{2}}}$,

we have

$y{\sqrt {1+y'^{2}}}-{\frac {yy'^{2}}{\sqrt {1+y'^{2}}}}=m$,

$m$ being the constant of integration, a quantity which will be considered more in detail later.

The above expression may be written

${\frac {y(1+y'^{2}-y'^{2})}{\sqrt {1+y'^{2}}}}=m$,

or

$y=m{\sqrt {1+y'^{2}}}\qquad {\text{[I]}}$.

From [I] it follows directly that

$y^{2}-m^{2}=m^{2}\left({\frac {{\text{d}}y}{{\text{d}}x}}\right)^{2}\qquad {\text{[II]}}$;

and [II], differentiated with respect to $x$, is

$y=m^{2}{\frac {{\text{d}}^{2}y}{{\text{d}}x^{2}}}$.

Two solutions of this diflFerential equation are

$y=e^{x/m}$ and $y=e^{-x/m}$,

so that the general solution is

$y=c_{1}e^{x/m}+c_{2}e^{-x/m}{\text{.}}\qquad {\text{[III]}}$

It appears that we have in this expression three arbitrary constants, $m$, $c_{1}$, and $c_{2}$; but from [II] we have, after substituting for $y^{2}$ and $\left({\frac {{\text{d}}y}{{\text{d}}x}}\right)^{2}$ their values from [III],

$m^{2}=4c_{1}c_{2}$.

Hence, writing in [III],

$c_{1}={\frac {1}{2}}me^{-x_{0}'/m}$ and $c_{2}={\frac {1}{2}}me^{x_{0}'/m}$

where $x_{0}'$ is a constant, we have

$y={\frac {1}{2}}m[e^{(x-x_{0}')/m}+e^{-(x-x_{0}')/m}]{\text{.}}\qquad {\text{[III']}}$

The two constants $x_{0}'$ and $m$ are determined from the two conditions that the curve is to pass through the two fixed points $P_{0}$ and $P_{1}$.

Article 31.
From what was given in Art. 19 it would appear that two neighboring curves are distinct throughout at least certain portions of their extent. This implies the existence of a certain neighborhood about the curve $C$ that is supposed to offer a minimum, within which this curve is not intersected by a neighboring curve. Suppose that the curve $C_{\epsilon }$ is derived from the curve $C$ by the substitution $y+\epsilon \eta$ for $y$ (cf. Art. 22). Consider the family of curves $(C_{\epsilon })$ obtained by varying $\epsilon$ between $-1$ and $+1$. For sufficiently small values of $\epsilon$ the curve $C_{\epsilon }$ will lie within the neighborhood presupposed to exist, and a portion of our family of curves will lie within this neighborhood. This 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. The fact that the surfaces generated by the curves $C_{\epsilon }$ are all greater than that generated by the curve $C$ does not prevent the existence of a neighboring curve constructed after a manner other than that by which the curves $C_{\epsilon }$ are produced, which would generate a surface of revolution having less surface area than that due to the revolution of $C$.

It is useful to determine for just what curve $C$ the above condition may be satisfied, and while this does not prove that the curve $C$ gives a minimal surface of revolution, it will at least limit the range of curves among which we may hope to find a generator of a minimal surface. Further investigation of this more limited range of curves may locate the curve or curves giving a minimal surface if such exists, and in the other case may prove their nonexistence. In the further investigation we shall derive the sufficient conditions to assure the existence of a maximum or a minimum.

Article 32.
The conclusions drawn from Art. 30 show that, if a curve exists which offers the required minimal surface, that curve must be a catenary. Since the catenary must pass through the two fixed points $P_{0}$ and $P_{1}$, we may determine the constants $m$ and $x_{0}'$ from the two relations (see formula [III'], Art. 30):

$y_{0}={\frac {1}{2}}m[e^{(x_{0}-x_{0}')/m}+e^{-(x_{0}-x_{0}')/m}]$

$y_{1}={\frac {1}{2}}m[e^{(x_{1}-x_{0}')/m}+e^{-(x_{1}-x_{0}')/m}]$

We shall see in the next Chapter that three cases arise according as the solution of the above equations furnish us with two catenaries, one catenary, or no catenary.

In the first place, it may be shown that the catenary nearest the $X$-axis can never furnish a minimal surface. The second case arises from the coincidence of the two catenaries just mentioned, and it will be seen that an infinite number of curves may in this case be drawn between the two points, each of which gives rise to the same rotation-area. These results are due to Todhunter (see references at the beginning of the next Chapter).