# Calculus of Variations/CHAPTER V

CHAPTER V: THE VARIATION OF CURVES EXPRESSED ANALYTICALLY. THE FIRST VARIATION.

• 74 General forms of the variations hitherto employed.
• 75 The functions ${\displaystyle \xi }$ and ${\displaystyle \eta }$. Their continuity.
• 76 Neighboring curves. The first variation.
• 77 The functions ${\displaystyle G}$, ${\displaystyle G_{1}}$ and ${\displaystyle G_{2}}$.
• 78 Proof of an important lemma.
• 79 The vanishing of the first variation and the differential equation ${\displaystyle G=0}$.
• 80 The curvature expressed in terms of ${\displaystyle F}$ and ${\displaystyle F_{1}}$.
• 81 The components ${\displaystyle w_{N}}$ and ${\displaystyle w_{T}}$ in the directions of the normal and the tangent.
• 82 Variations in the direction of the tangent and in the direction of the normal.
• 83 Discontinuities in the path of integration. Irregular curves.
• 84,85 Problem of Euler illustrating the preceding article.
• 86 Summary.

Article 74.
In Chapter II we considered examples of special variations. The method followed provided for the displacement of a curve in one direction only, in the direction parallel to the ${\displaystyle X}$-axis, and is consequently applicable only to the comparison of integrals along curves obtained from one another by such a deformation.

We shall now give a more general form to the variations employed and shall seek strenuous methods for the solution of the general problem of variations proposed in Chapter I. After deriving the necessary conditions we shall then proceed to discuss the sufficient conditions. In order to develop the conditions for the appearance of a maximum or a minimum of the integral

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

it is necessary to study more closely the conception of the variation of a curve and fix this conception analytically.

By writing instead of each point ${\displaystyle x}$, ${\displaystyle y}$ of a curve (presupposed regular) another point ${\displaystyle x+\xi }$, ${\displaystyle y+\eta }$, we transform the first curve into another regular curve. This second curve is neighboring the first curve if we make sufficiently small the quantities ${\displaystyle \xi }$ and ${\displaystyle \eta }$ which like ${\displaystyle x}$ and ${\displaystyle y}$ we consider as one-valued continuous functions of ${\displaystyle t}$.

Article 75.
The following is one of the methods of effecting this result. Let ${\displaystyle \xi }$ and ${\displaystyle \eta }$ be continuous functions of ${\displaystyle t}$ and also of a quantity ${\displaystyle k}$. We further suppose that ${\displaystyle \xi }$ and ${\displaystyle }$\eta vanish when ${\displaystyle k=0}$ for ever value of ${\displaystyle t}$, for example

${\displaystyle \xi =ku(t)~;~\eta =kv(t)}$,

${\displaystyle u}$ and ${\displaystyle v}$ being finite and continuous functions of ${\displaystyle t}$.

The functions ${\displaystyle u}$ and ${\displaystyle v}$ and consequently also ${\displaystyle \xi }$ and ${\displaystyle \eta }$ are subject to further conditions. It is in general required to construct a curve between two given points which first may be regarded as fixed. Later the condition of their variability may be introduced. Consequently we have to consider only such values of ${\displaystyle t}$ that ${\displaystyle \xi }$, ${\displaystyle \eta }$ and consequently ${\displaystyle u}$, ${\displaystyle v}$ vanish on the limits.

If for ${\displaystyle x}$, ${\displaystyle y}$ we write ${\displaystyle x+\xi }$, ${\displaystyle y+\eta }$, then for ${\displaystyle x'}$, ${\displaystyle y'}$ we must write ${\displaystyle x'+\xi '}$, ${\displaystyle y'+\eta '}$. Further, the function ${\displaystyle F(x+\xi ,y+\eta ,x'+\xi ',y'+\eta ')}$ must be developed in powers of ${\displaystyle \xi }$, ${\displaystyle \eta }$, ${\displaystyle \xi '}$ and ${\displaystyle \eta '}$. For the convergence of this series, it is necessary that ${\displaystyle \xi }$, ${\displaystyle \eta }$, ${\displaystyle \xi '}$ and ${\displaystyle \eta '}$ have finite values.

Now, if we write

${\displaystyle \xi =k\sin {\frac {t}{k^{n}}}~;~\eta =k\cos {\frac {t}{k^{n}}}}$,

then we have

${\displaystyle {\frac {{\text{d}}\xi }{{\text{d}}t}}=k^{1-n}\cos {\frac {t}{k^{n}}}~;~{\frac {{\text{d}}\eta }{{\text{d}}t}}=-k^{1-n}\sin {\frac {t}{k^{n}}}}$,

so that, whereas ${\displaystyle \xi }$ and ${\displaystyle \eta }$ have infinitely small values for infinitely small values of ${\displaystyle k}$, the quantities ${\displaystyle {\frac {{\text{d}}\xi }{{\text{d}}t}}}$, ${\displaystyle {\frac {{\text{d}}\eta }{{\text{d}}t}}}$ vacillate for ${\displaystyle n-1}$ between ${\displaystyle +1}$ and ${\displaystyle -1}$ and become infinite for ${\displaystyle n>1}$. We shall consequently consider only such special variations in which ${\displaystyle u}$ and ${\displaystyle v}$ are functions of ${\displaystyle t}$ alone, and which with their derivatives are finite and continuous between the limits ${\displaystyle t_{0}}$ and ${\displaystyle t_{1}}$. We thus restrict, in a great measure, the arbitrariness of the indefinitely small variations of the curve, and thus exclude a great many neighboring curves from the discussion. However, there exist among all the possible neighboring curves also such which satisfy the above conditions, and with these we shall first establish the necessary conditions and later show that the necessary conditions thus established are also sufficient for the establishment of the existence of a maximum or a minimum value of the integral. (See Arts. 134 et seq.)

Article 76.
We have instead of one neighboring curve a whole bundle of such curves if we make the substitutions

${\displaystyle x\rightarrow x+\epsilon \xi }$, ${\displaystyle y\rightarrow y+\epsilon \eta }$, ${\displaystyle x'\rightarrow x'+\epsilon \xi '}$, ${\displaystyle y'\rightarrow y'+\epsilon \eta '}$,

and let ${\displaystyle \epsilon }$, a quantity independent of the variables in ${\displaystyle F(x,y,x',y')}$, vary between ${\displaystyle +1}$ and ${\displaystyle -1}$.

The total variation that is thereby introduced in the integral of the preceding article is

${\displaystyle I+\Delta I=\in _{t_{0}}^{t_{1}}F(x+\epsilon \xi ,y+\epsilon \eta ,x'+\epsilon \xi ',y'+\epsilon \eta ')~{\text{d}}t}$,

which developed by Maclaurin's Theorem is

${\displaystyle =\int _{t_{0}}^{t_{1}}\left(F+\epsilon \left[{\frac {\partial F}{\partial x}}\xi +{\frac {\partial F}{\partial y}}\eta +{\frac {\partial F}{\partial x'}}\xi '+{\frac {\partial F}{\partial y'}}\eta '\right]+\epsilon ^{2}(\cdots )\right)~{\text{d}}t}$.

Further (see Art. 25)

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

hence, equating coefficients of ${\displaystyle \epsilon }$,

${\displaystyle 2)\qquad \delta I=\int _{t_{0}}^{t_{1}}\left[{\frac {\partial F}{\partial x}}\xi +{\frac {\partial F}{\partial y}}\eta +{\frac {\partial F}{\partial x'}}\xi '+{\frac {\partial F}{\partial y'}}\eta '\right]~{\text{d}}t}$.

But

${\displaystyle \int _{t_{0}}^{t_{1}}{\frac {\partial F}{\partial x'}}{\frac {{\text{d}}\xi }{{\text{d}}t}}~{\text{d}}t=\left[{\frac {\partial F}{\partial x'}}\xi \right]_{t_{0}}^{t_{1}}-\int _{t_{0}}^{t_{1}}\xi {\frac {\text{d}}{{\text{d}}t}}\left({\frac {\partial F}{\partial x'}}\right)~{\text{d}}t}$;

So that 2) becomes

${\displaystyle 2')\qquad \delta I=\int _{t_{0}}^{t_{1}}\left(\left[{\frac {\partial F}{\partial x}}-{\frac {\text{d}}{{\text{d}}t}}\left({\frac {\partial F}{\partial x'}}\right)\right]\xi +\left[{\frac {\partial F}{\partial y}}-{\frac {\text{d}}{{\text{d}}t}}\left({\frac {\partial F}{\partial y'}}\right)\right]\eta \right)~{\text{d}}t+\left[{\frac {\partial F}{\partial x'}}\xi +{\frac {\partial F}{\partial y'}}\eta \right]_{t_{0}}^{t_{1}}}$;

or

${\displaystyle 3)\qquad \delta I=\int _{t_{0}}^{t_{1}}\left(G_{1}\xi +G_{2}\eta \right)~{\text{d}}t+\left[{\frac {\partial F}{\partial x'}}\xi +{\frac {\partial F}{\partial y'}}\eta \right]_{t_{0}}^{t_{1}}}$,

where

${\displaystyle G_{1}={\frac {\partial F}{\partial x}}-{\frac {\text{d}}{{\text{d}}t}}\left({\frac {\partial F}{\partial x'}}\right)}$, ${\displaystyle G_{2}={\frac {\partial F}{\partial y}}-{\frac {\text{d}}{{\text{d}}t}}\left({\frac {\partial F}{\partial y'}}\right)}$.

Article 77.
Owing to the hypotheses that ${\displaystyle x'}$, ${\displaystyle y'}$, ${\displaystyle {\frac {{\text{d}}\xi }{{\text{d}}t}}}$, ${\displaystyle {\frac {{\text{d}}\eta }{{\text{d}}t}}}$ vary in a continuous manner with ${\displaystyle t}$ [that is, within the portion of curve considered no sudden change enters in the direction of the curve] , the first variation of the integral ${\displaystyle I}$ may be transformed in a remarkable manner.

${\displaystyle >G_{1}={\frac {\partial F}{\partial x}}-{\frac {\text{d}}{{\text{d}}t}}\left({\frac {\partial F}{\partial x'}}\right)}$

and also (Art. 72)

${\displaystyle F(x,y,x',y')=x'{\frac {\partial F}{\partial x'}}+y'{\frac {\partial F}{\partial y'}}}$.

Therefore

${\displaystyle {\frac {\partial F}{\partial x}}=x'{\frac {\partial ^{2}F}{\partial x'\partial x}}+y'{\frac {\partial ^{2}F}{\partial y'\partial x}}}$;

and differentiating ${\displaystyle {\frac {\partial F}{\partial x'}}}$ with respect to ${\displaystyle t}$, we have

${\displaystyle -{\frac {\text{d}}{{\text{d}}t}}\left({\frac {\partial F}{\partial x}}\right)=-{\frac {\partial ^{2}F}{\partial x\partial x'}}{\frac {{\text{d}}x}{{\text{d}}t}}-{\frac {\partial ^{2}F}{\partial y\partial x'}}{\frac {{\text{d}}y}{{\text{d}}t}}-{\frac {\partial ^{2}F}{\partial x'^{2}}}{\frac {{\text{d}}x'}{{\text{d}}t}}-{\frac {\partial ^{2}F}{\partial y'\partial x'}}{\frac {{\text{d}}y'}{{\text{d}}t}}}$.

Hence,

${\displaystyle G_{1}=y'\left({\frac {\partial ^{2}F}{\partial x\partial y'}}-{\frac {\partial ^{2}F}{\partial y\partial x'}}\right)-\left({\frac {\partial ^{2}F}{\partial x'^{2}}}{\frac {{\text{d}}x'}{{\text{d}}t}}+{\frac {\partial ^{2}F}{\partial y'\partial x'}}{\frac {{\text{d}}y'}{{\text{d}}t}}\right)}$.

Writing ${\displaystyle {\frac {\partial ^{2}F}{\partial x'^{2}}}=y'^{2}F_{1}}$, (Art. 73 ), and ${\displaystyle {\frac {\partial ^{2}F}{\partial y'\partial x'}}=-x'y'F_{1}}$, and defining ${\displaystyle G}$ by the equation

${\displaystyle G={\frac {\partial ^{2}F}{\partial x\partial y'}}-{\frac {\partial ^{2}F}{\partial y\partial x'}}-F_{1}\left(y'{\frac {{\text{d}}x'}{{\text{d}}t}}-x'{\frac {{\text{d}}y'}{{\text{d}}t}}\right)}$,

it is seen that

${\displaystyle G_{1}=y'G}$.

In a similar manner it may be shown that

${\displaystyle G_{2}=-x'G}$.

Article 7.
Lemma. If ${\displaystyle \phi (t)}$ and ${\displaystyle \psi (t)}$ are two continuous functions of ${\displaystyle t}$ between the limits ${\displaystyle t_{0}}$ and ${\displaystyle t_{1}}$ and if the integral

${\displaystyle \int _{t_{0}}^{t_{1}}\phi (t)\psi (t)~{\text{d}}t}$

is always zero, in whatever manner ${\displaystyle \psi (t)}$ is chosen, then necessarily ${\displaystyle \phi (t)}$ must vanish for every value of ${\displaystyle t}$ between ${\displaystyle t_{0}}$ and ${\displaystyle t_{1}}$.

The following proof, due to Prof. Schwarz, is a geometrical interpretation of a method due to Heine.[1]

Suppose it possible that the function ${\displaystyle \phi (t)}$ has a finite value for a point ${\displaystyle t=t'}$ situated between ${\displaystyle t_{0}}$ and ${\displaystyle t_{1}}$. Then owing to the continuity of ${\displaystyle \phi (t)}$ we can find an interval ${\displaystyle t'-d\ldots t'+d}$ within which ${\displaystyle \phi (t)}$ also has a finite value.

We write the integral in the form

${\displaystyle \int _{t_{0}}^{t_{1}}\phi (t)\psi (t)~{\text{d}}t=\int _{t_{0}}^{t'-d}\phi (t)\psi (t)~{\text{d}}t+\int _{t'-d}^{t'+d}\phi (t)\psi (t)~{\text{d}}t+\int _{t'+d}^{t_{1}}\phi (t)\psi (t)~{\text{d}}t}$.

The second integral on the right hand side may be written

${\displaystyle M=\int _{t'-d}^{t'+d}\psi (t)~{\text{d}}t}$,

where ${\displaystyle M}$ is the mean value of ${\displaystyle \phi (t)}$ for a value of ${\displaystyle t}$ within the interval ${\displaystyle t'-d\ldots t'+d}$.

We shall show that it is possible to determine a function ${\displaystyle \psi (t)}$ which will render this integral positive and greater than the sum of the first and third integrals in the above expression, while at the same time ${\displaystyle \psi (t_{0})=psi(t_{1})=0}$.

Let us form the equation

${\displaystyle \left(t-1+{\frac {x^{2}}{d^{2}}}\right)y=0}$,

which represents the parabola ${\displaystyle y=1-{\frac {x^{2}}{d^{2}}}}$ and the ${\displaystyle X}$-axis.

Consider next the equation

${\displaystyle \left(t-1+{\frac {x^{2}}{d^{2}}}\right)y=\epsilon ^{2}}$,

where ${\displaystyle \epsilon }$ is a small quantity. By taking ${\displaystyle \epsilon }$ sufficiently small, this curve can be made to approach as near as we wish the parabola and the ${\displaystyle X}$-axis.

Solving the above equation for ${\displaystyle y}$, we have as the two roots (two branches)

${\displaystyle y={\frac {1}{2}}\left(1-{\frac {x^{2}}{d^{2}}}\right)\pm {\sqrt {{\frac {1}{4}}\left(1-{\frac {x^{2}}{d^{2}}}\right)^{2}+\epsilon ^{2}}}}$.

The branch

${\displaystyle y={\frac {1}{2}}\left(1-{\frac {x^{2}}{d^{2}}}\right)+{\sqrt {{\frac {1}{4}}\left(1-{\frac {x^{2}}{d^{2}}}\right)^{2}+\epsilon ^{2}}}}$

is symmetrical with respect to the ${\displaystyle Y}$-axis, and for values of ${\displaystyle x}$, such that ${\displaystyle -d\leq x\leq +d}$, the ordinate of any point of the curve is greater than the corresponding ordinate of the parabola.

For the parabola ${\displaystyle y=1-{\frac {x^{2}}{d^{2}}}}$, the integral

${\displaystyle \int _{-d}^{+d}y~{\text{d}}x={\frac {4}{3}}d}$

It follows then that for the curve we must have

${\displaystyle \int _{-d}^{+d}\left[frac{1}{2}\left(1-{\frac {x^{2}}{d^{2}}}\right)+{\sqrt {{\frac {1}{4}}\left(1-{\frac {x^{2}}{d^{2}}}\right)^{2}+\epsilon ^{2}}}\right]~{\text{d}}x>{\frac {4}{3}}d}$;

for ${\displaystyle |x|=d}$, we have ${\displaystyle y=\epsilon }$; and from the inequality

${\displaystyle y=-frac{1}{2}\left(1-{\frac {x^{2}}{d^{2}}}\right)+{\sqrt {{\frac {1}{4}}\left(1-{\frac {x^{2}}{d^{2}}}\right)^{2}+\epsilon ^{2}}}<-frac{1}{2}\left(1-{\frac {x^{2}}{d^{2}}}\right)+{\sqrt {{\frac {1}{4}}\left(1-{\frac {x^{2}}{d^{2}}}\right)^{2}+\epsilon }}}$,

it follows for ${\displaystyle |x|, that ${\displaystyle y}$ is positive and ${\displaystyle <\epsilon }$. For the lower branch ${\displaystyle y}$ is negative and the curve

${\displaystyle y=-frac{1}{2}\left(1-{\frac {x^{2}}{d^{2}}}\right)-{\sqrt {{\frac {1}{4}}\left(1-{\frac {x^{2}}{d^{2}}}\right)^{2}+\epsilon ^{2}}}}$

follows the parabola to infinity as shown in the figure. It is however the upper branch which we use, since ${\displaystyle y}$ is less than ${\displaystyle \epsilon }$, as soon as ${\displaystyle x}$ passes the value ${\displaystyle d}$ on either side of the origin.

Instead of the integral last written, take the integral which has the same value

${\displaystyle \int _{t'-d}^{t'+d}\left({\frac {1}{2}}\left(1-{\frac {(t-t')^{2}}{d^{2}}}\right)+{\sqrt {{\frac {1}{4}}\left(1-{\frac {(t-t')^{2}}{d^{2}}}\right)^{2}+\epsilon ^{2}}}\right)~{\text{d}}t>{\frac {4}{3}}d}$.

Writing

${\displaystyle \chi (t)={\frac {1}{2}}\left(1-{\frac {(t-t')^{2}}{d^{2}}}\right)+{\sqrt {{\frac {1}{4}}\left(1-{\frac {(t-t')^{2}}{d^{2}}}\right)^{2}+\epsilon ^{2}}}}$,

we have

${\displaystyle \int _{t_{0}}^{t_{1}}\phi (t)\chi (t)~{\text{d}}t=\int _{t_{0}}^{t'-d}\phi (t)\chi (t)~{\text{d}}t+\int _{t'-d}^{t'+d}\phi (t)\chi (t)~{\text{d}}t+\int _{t'+d}^{t_{1}}\phi (t)\chi (t)~{\text{d}}t}$
${\displaystyle =M'\int _{t_{0}}^{t'-d}\chi (t)~{\text{d}}t+M\int _{t'-d}^{t'+d}\chi (t)~{\text{d}}t+M''\int _{t'+d}^{t_{1}}\chi (t)~{\text{d}}t}$
${\displaystyle =M'[<\epsilon ](t'-d-t_{0})+M\left[>{\frac {4}{3}}d\right]+M''[<\epsilon ](t_{1}-t'-d)}$,

where ${\displaystyle M'}$, ${\displaystyle M}$ and ${\displaystyle M''}$ are mean values of ${\displaystyle \phi (t)}$ in the respective intervals and where ${\displaystyle [<\epsilon ]}$ denotes that the quantity that stands within the brackets is less than ${\displaystyle \epsilon }$.

It is seen that by taking ${\displaystyle \epsilon }$ sufficiently small that the sign of the integral ${\displaystyle \int _{t_{0}}^{t_{1}}\phi (t)\chi (t)~{\text{d}}t}$ is determined by that of ${\displaystyle M~{\frac {4}{3}}d}$and consequently this integral is different form zero.

Instead of the function ${\displaystyle \chi (t)}$, write

${\displaystyle \phi (t)=\left({\frac {t-t_{0}}{t'-t_{0}}}\right)^{m}\left({\frac {t_{1}-t}{t_{1}-t'}}\right)^{n}\chi (t)}$,

where ${\displaystyle m}$ and ${\displaystyle n}$ are positive integers.

We see that ${\displaystyle \phi (t_{0})=0=\phi (t_{1})}$, and as above, it follows that

${\displaystyle \int _{t_{0}}^{t_{1}}\phi (t)\psi (t)~{\text{d}}t\neq 0}$.

Hence, on the supposition that ${\displaystyle \phi (t)\neq 0}$ for a point of the curve alonjj which we integrate, it follows that a function ${\displaystyle \psi (t)}$ can be found which causes the above integral to be different from zero.

But as this integral was supposed to be zero for all functions ${\displaystyle \psi (t)}$, it follows that we must have ${\displaystyle \phi (t)=0}$ for all values of ${\displaystyle t}$ between ${\displaystyle t_{0}}$ and ${\displaystyle t_{1}}$.

Article 79.
In the expression

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

unless <math\delta I[/itex] and ${\displaystyle \epsilon }$ always retain the same sign, it is necessary that ${\displaystyle \delta I}$ be zero in order that ${\displaystyle \Delta I}$ be continuously negative or continuously positive; i.e., in order that the integral ${\displaystyle I}$ be a maximum or a minimum (see Art. 26).

Substitute for ${\displaystyle G_{1}}$ and ${\displaystyle G_{2}}$ their values in terms of ${\displaystyle G}$ from Art. 77, in the expression for ${\displaystyle \delta I}$ of Art. 76, and we have

${\displaystyle \delta I=\int _{t_{0}}^{t_{1}}G(t'\xi -x'\eta )~{\text{d}}t+\left[\xi {\frac {\partial F}{\partial x'}}+\eta {\frac {\partial F}{\partial y'}}\right]_{t_{0}}^{t_{1}}}$.

If we suppose that the points ${\displaystyle P_{0}}$ and ${\displaystyle P_{1}}$ are fixed so that ${\displaystyle \xi =0=\eta }$ for them, then the boundary terms vanish. Further, since ${\displaystyle y'\xi -x'\eta }$ is an arbitrary and continuous function of ${\displaystyle t}$, it follows from the above lemma that, in order for ${\displaystyle \delta t}$ to be zero, ${\displaystyle G=0}$ for every point of the curve within the interval ${\displaystyle t_{0}\ldots t_{0}}$. ${\displaystyle G}$ can not have a finite value different from zero for isolated points on the curve, since this portion of curve must be continuous in order that the integral may have a meaning.

The differential equation ${\displaystyle G=0}$ of the second order is a necessary condition for a maximum or a minimum value of ${\displaystyle I}$, and will afford the required curve if such curve exists. We note that it is independent of the manner of variation, as the quantities ${\displaystyle \xi }$ and ${\displaystyle \eta }$ do not appear in it.

From the relation (see Art. 77)

${\displaystyle \delta I=\int _{t_{0}}^{t_{1}}(G_{1}\xi +G_{2}\eta )~{\text{d}}t=0}$,

it follows that

${\displaystyle \int _{t_{0}}^{t_{1}}G_{1}\xi ~{\text{d}}t+\int _{t_{0}}^{t_{1}}G_{2}\eta ~{\text{d}}t=0}$.

Among all possible variations there are those for which ${\displaystyle \eta =0}$, and consequently

${\displaystyle \int _{t_{0}}^{t_{1}}G_{1}\xi ~{\text{d}}t=0}$.

As above we have then

${\displaystyle G_{1}=0}$,

and similarly

${\displaystyle G_{2}=0}$.

Further, if we multiply ${\displaystyle y'G=G_{1}}$ and ${\displaystyle -x'G=G_{2}}$ respectively by ${\displaystyle v'}$ and ${\displaystyle -x'}$, we have by addition

${\displaystyle (y'^{2}+x'^{2})G=0}$,

and as ${\displaystyle x'}$ and ${\displaystyle y'}$ cannot both vanish simultaneously, it follows that ${\displaystyle G=0}$. Hence the equations ${\displaystyle G_{1}=0}$, ${\displaystyle G_{2}=0}$ on the one hand, and ${\displaystyle G=0}$ on the other, are necessary consequences of one another.

The equations ${\displaystyle G_{1}=0=G_{2}}$ often more convenient than ${\displaystyle G=0}$; especially is this the case if the function ${\displaystyle F}$ does not contain explicitly one of the two quantities ${\displaystyle x}$ and ${\displaystyle y}$. If ${\displaystyle x}$, for instance, is wanting, then ${\displaystyle {\frac {\partial F}{\partial x}}=0}$, and from

${\displaystyle G_{1}={\frac {\partial F}{\partial x}}-{\frac {\text{d}}{{\text{d}}t}}{\frac {\partial F}{\partial x'}}=0}$,

it follows that ${\displaystyle {\frac {\partial F}{\partial x'}}=}$ constant.

Article 80.
The curvature at any point of a curve is denoted by

${\displaystyle \kappa ={\frac {1}{\rho }}={\frac {x'y''-y'x''}{[x'^{2}y'^{2}]^{3/2}}}}$,

and owing to the equation

${\displaystyle G=0={\frac {\partial ^{2}F}{\partial y\partial x'}}-{\frac {\partial ^{2}F}{\partial x\partial y'}}-F(x'y''-x''y')}$,

we have

${\displaystyle {\frac {1}{\rho }}={\frac {{\frac {\partial ^{2}F}{\partial y\partial x'}}-{\frac {\partial ^{2}F}{\partial x\partial y'}}}{[x'^{2}+y'^{2}]^{3/2}F_{1}}}}$,

an expression which depends upon ${\displaystyle x}$,${\displaystyle y}$,${\displaystyle x'}$,${\displaystyle y'}$ alone and not upon the higher derivatives.

It is thus seen that through the equation ${\displaystyle G=0}$, a definite relation is expressed between the curvature of a curve at a point, the coordinates and the direction of the tangent of the curve at this point.

Article 81.
Let a point ${\displaystyle P}$ on the curve be transformed by a variation into the point ${\displaystyle P'}$ and let the displacement ${\displaystyle pp'}$ be denoted by ${\displaystyle v}$; further, let the components in the ${\displaystyle x}$ and ${\displaystyle y}$ directions be ${\displaystyle \xi }$ and ${\displaystyle \eta }$, while ${\displaystyle w_{N}}$ and ${\displaystyle w_{T}}$ denote the components of this displacement in the direction of the normal and the tangent to the curve at the point ${\displaystyle P}$.

Let ${\displaystyle \lambda }$ denote the angle between these directions and let the direction cosines of the normal be denoted by

${\displaystyle {\frac {{\text{d}}x}{\sqrt {\left({\frac {{\text{d}}x}{{\text{d}}t}}\right)^{2}+\left({\frac {{\text{d}}y}{{\text{d}}t}}\right)^{2}}}}}$ and ${\displaystyle {\frac {{\text{d}}y}{\sqrt {\left({\frac {{\text{d}}x}{{\text{d}}t}}\right)^{2}+\left({\frac {{\text{d}}y}{{\text{d}}t}}\right)^{2}}}}}$; i.e., by ${\displaystyle {\frac {x'}{s'}}}$ and ${\displaystyle {\frac {y'}{s'}}}$.

Then from analytical geometry,

${\displaystyle w_{T}=\xi \cos(\lambda )+\eta \sin(\lambda )={\frac {x'\xi +y'\eta }{\sqrt {x'^{2}+y'^{2}}}}}$,

and therefore

${\displaystyle w_{N}=\eta \cos(\lambda )-\xi \sin(\lambda )={\frac {x'\eta -y'\xi }{\sqrt {x'^{2}+y'^{2}}}}}$;

and

${\displaystyle w_{T}^{2}+w_{N}^{2}=\xi ^{2}+\eta ^{2}}$.

These expressions substituted in the formula for ${\displaystyle \delta I}$ (Art. 79) give

${\displaystyle \delta I=-\int _{t_{0}}^{t_{1}}w_{N}G~{\text{d}}s+\left[{\frac {Fw_{T}}{\sqrt {x'^{2}+y'^{2}}}}+{\frac {x'{\frac {\partial F}{\partial y'}}-y'{\frac {\partial F}{\partial x'}}}{\sqrt {x'^{2}+y'^{2}}}}\right]_{t_{0}}^{t_{1}}}$.

From this it is seen that only the component of the variation which is in the direction of the normal enters under the integral sign.

Article 82.
By means of the formula below we will prove that the variation in the direction of the tangent brings forth only such terms for the first variation that are free from, the sign of integration.

As in Art. 76, write

${\displaystyle \delta I=\int _{t_{0}}^{t_{1}}\left({\frac {\partial F}{\partial x}}\xi +{\frac {\partial F}{\partial x'}}\xi '+{\frac {\partial F}{\partial y}}\eta +{\frac {\partial F}{\partial y'}}\eta '\right)~{\text{d}}t}$,

and, substituting in this expression ${\displaystyle \xi =v{\frac {{\text{d}}x}{{\text{d}}s}}}$, ${\displaystyle \eta =v{\frac {{\text{d}}y}{{\text{d}}s}}}$, we have

${\displaystyle \delta I=\int _{t_{0}}^{t_{1}}\left[v\left({\frac {\partial F}{\partial x}}{\frac {{\text{d}}x}{{\text{d}}s}}+{\frac {\partial F}{\partial y}}{\frac {{\text{d}}y}{{\text{d}}s}}\right)+{\frac {\partial F}{\partial x'}}{\frac {\text{d}}{{\text{d}}t}}\left(v{\frac {{\text{d}}x}{{\text{d}}s}}\right)+{\frac {\partial F}{\partial y'}}{\frac {\text{d}}{{\text{d}}t}}\left(v{\frac {{\text{d}}y}{{\text{d}}s}}\right)\right]~{\text{d}}t}$.

Noting that

${\displaystyle \int _{t_{0}}^{t_{1}}{\frac {\partial F}{\partial x'}}{\frac {\text{d}}{{\text{d}}t}}\left(v{\frac {{\text{d}}x}{{\text{d}}s}}\right)~{\text{d}}t=\left[{\frac {\partial F}{\partial x'}}v{\frac {{\text{d}}x}{{\text{d}}s}}\right]_{t_{0}}^{t_{1}}-\int _{t_{0}}^{t_{1}}v{\frac {{\text{d}}x}{{\text{d}}s}}{\frac {\text{d}}{{\text{d}}t}}\left({\frac {\partial F}{\partial x'}}\right)~{\text{d}}t}$,

it is seen that

${\displaystyle \delta I=\int _{t_{0}}^{t_{1}}\left(v{\frac {{\text{d}}x}{{\text{d}}s}}\left[{\frac {\partial F}{\partial x}}-{\frac {\text{d}}{{\text{d}}t}}\left({\frac {\partial F}{\partial x'}}\right)\right]+v{\frac {{\text{d}}y}{{\text{d}}s}}\left[{\frac {\partial F}{\partial y}}-{\frac {\text{d}}{{\text{d}}t}}\left({\frac {\partial F}{\partial y'}}\right)\right]\right)~{\text{d}}t+\left[{\frac {\partial F}{\partial x'}}v{\frac {{\text{d}}x}{{\text{d}}s}}+{\frac {\partial F}{\partial y'}}v{\frac {{\text{d}}y}{{\text{d}}s}}\right]_{t_{0}}^{t_{1}}}$
${\displaystyle =\int _{t_{0}}^{t_{1}}G(y'\xi -x'\eta )~{\text{d}}t+\left[{\frac {\partial F}{\partial x'}}v{\frac {{\text{d}}x}{{\text{d}}s}}+{\frac {\partial F}{\partial y'}}v{\frac {{\text{d}}y}{{\text{d}}s}}\right]_{t_{0}}^{t_{1}}}$.

But

${\displaystyle t'\xi -x'\eta =y'v{\frac {{\text{d}}x}{{\text{d}}s}}-x'v{\frac {{\text{d}}y}{{\text{d}}s}}={\frac {v}{{\text{d}}s}}\left[y'{\frac {{\text{d}}x}{{\text{d}}t}}-x'{\frac {{\text{d}}y}{{\text{d}}t}}\right]{\text{d}}t\equiv 0}$,

so that everything under the sign of integration drops out, leaving

${\displaystyle \delta I=\left[{\frac {\partial F}{\partial x'}}v{\frac {{\text{d}}x}{{\text{d}}s}}+{\frac {\partial F}{\partial y'}}v{\frac {{\text{d}}y}{{\text{d}}s}}\right]_{t_{0}}^{t_{1}}}$.

Hence, if we make a sliding of the curve by the substitution

${\displaystyle x\rightarrow x+\epsilon \xi \qquad y\rightarrow y+\epsilon \eta }$,

and resolve this sliding into two components, of which the one is parallel to the direction of the tangent and the other is parallel to the direction of the normal, then the result of the sliding in the direction of the tangent is seen only in the terms which have reference to the limits, and all these terms are exact differentials under the sign of integration, while the effect due to a sliding in the direction of the normal is shown in the formula of the preceding article.

Article 83.
The expression for the first variation has been obtained on the hypothesis that the elements of integration have for each point of the path of integration a one-valued meaning. In case the path involved, discontinuities, it could be resolved into a finite number of portions of regular curve, and along each portion ${\displaystyle \delta I}$ would have a meaning similar to that of the preceding article. It is assumed, then, that the required curve, which is to furnish a maximum or a minimum value of the integral, is regular in its whole trace, or, at least, that it consists of regular portions of curve. In the latter case we shall at first limit ourselves to the consideration of one such portion. Within this portion of curve not only ${\displaystyle x}$,${\displaystyle y}$ but also ${\displaystyle x'}$,${\displaystyle y'}$ will be one-valued functions of ${\displaystyle t}$. This assumption is already implicitly contained in the assumption of the possibility of the development of ${\displaystyle \Delta I}$ by Taylor's Theorem; for otherwise the derivatives of ${\displaystyle F}$ according to ${\displaystyle x'}$ and ${\displaystyle y'}$ for the curve which has to be developed could not be formed.

That these assumptions have been made is due to the fact that otherwise the curve could not be the object of a mathematical investigation, since there are no methods of representing irregular curves in their entire generality. If therefore one is contented with the rules of differentiation and integration, he must extend these considerations over only such functions to which the rules may be applied, that is, to the functions having the properties above. There are many problems in geometry and mechanics for which the above hypotheses cannot be made.

Article 84.
The following problem proposed by Euler illustrates what has just been said:

Required a curve connecting two fixed points such that the area between the curve, its evolute and the radii of curvature at its extremities may be a minimum.

The analytical solution of this problem is the arc of a cycloid, if indeed there exists a minimum. We shall now show that such is not the case. For join the two fixed points ${\displaystyle A}$ and ${\displaystyle B}$ by a straight line which divide into ${\displaystyle n}$ equal parts, and draw alternately

above and below the line ${\displaystyle n}$ semi-circles having the ${\displaystyle n}$ parts of the line as diameters.

All the radii of curvature of each semi-circle, i.e., of each portion of curve which is to be a minimum, intersect on the line ${\displaystyle AB}$ and it is evident that

${\displaystyle {\frac {n}{2}}\pi \left({\frac {AB}{2n}}\right)^{2}={\frac {\pi {\overline {AB}}^{2}}{8n}}}$

must be a minimum.

If we increase the number ${\displaystyle n}$ sufficiently, we may make the above expression become arbitrarily small; and in the limit ${\displaystyle n=\infty }$ the curve will tend to become the straight line ${\displaystyle AB}$. From this it is evident that there is not present a minimum surface area.

Article 85.
The same result would have been obtained, if instead of the straight line ${\displaystyle AB}$ we had taken the arc of a cycloid through these points, and had then drawn a system of small cycloids having their cusps along the large cycloid. (See Todhunter, Researches in the Calculus of Variations, p. 252.) The reason that a minimum is not given through the large cycloid is due to the fact that such a minimum is ofiEered by an irregular curve, and that this irregular curve is not included in our analytical research.[2] It follows that our assumption made regarding the regularity of the curve is out of place and leads to something untrue.

But in spite of the not improbable possibility that the curve which is to satisfy a given proposition is irregular, we must make the hypothesis that the curve is regular, since we come to analytical difEerential equations only by limiting our investigations to such regular curves, and the most general theory of functions teaches that in turn through these diflEerential equations are defined the analytical functions which in their whole extent have existing derivatives.

Article 86.
To avoid any misunderstanding, we repeat what we have already said in the previous Chapter: it is not asserted that there is anything in the nature of the problem whereby one may a priori conclude that the required curve must be regular. Having these hypotheses, we fix our ideas and draw deductions. After the solution of the problem has been effected, we have to make in addition a special proof that the derived curve has all the required properties, and that this curve is the only one which has them.

The chief difficulty in all such problems, as we have shown above in the special problem of approximation (or of the passing to a limit), consists in showing that the regular curve that has been found, found indeed from the necessary conditions, also at the same time satisfies the sufficient conditions, and therefore satisfies all the requirements of the problem.

1. Heine, Crelle, bd. 54, p. 338.