Calculus of Variations/CHAPTER IX

CHAPTER IX: CONJUGATE POINTS.

• 122 The second variation of the differential equation $J=0$.
• 123,124 The solutions of the equations $G=0$ and $J=0$. The second variation derived from the first variation.
• 125 Variations of the constants in the solutions of $G=0$.
• 126 The solutions $u_1$ and $u_2$ of the differential equation $J=0$.
• 127 These solutions are independent of each other.
• 128 The function $\mathrm{\Theta }(t,t^{\prime })$. Conjugate points.
• 129 The relative position of conjugate points on a curve.
• 130 Graphical representation of the ratio $\frac{u_1}{u_2}$.
• 131 Summary.
• 132 Points of intersection of the curves $G=0$ and $\delta G=0$.
• 133 The second variation when two conjugate points are the limits of integration, and when a pair of conjugate points are situated between these limits.

Article 122.
The condition given in the preceding Chapter is not sufficient to establish the existence of a maximum or a minimum. Under the assumption that $F_1$ is neither zero nor infinite within the interval $t_0\dots t_1$, suppose that two functions ${\varphi }_1(t)$ and ${\varphi }_2(t)$ can be found which satisfy the differential equation 13) of the last Chapter, so that, consequently,

$u=c_1{\varphi }_1(t)+c_2{\varphi }_2(t)$

is the general solution of $J=0$. Then, even if within the limits of integration it can be shown that $u$ is not infinite, it may still happen that, however the constants $c_1$ and $c_2$ be chosen, the function $u$ vanishes, so that the transformation of the $v$-equation into the $u$-equation is not admissible ; consequently nothing can be determined regarding the appearance of a maximum or a minimum. We are thus led again to the necessity of studying more closely the function $u$ defined by the equation $J=0$, in order that we may determine under what conditions this function does not vanish within the interval $t_0\dots t_1$.

It is seen that the equation $J=0$ is satisfied, if for $u$ we write

$u_1=-F_1{u}^{\prime }$ [see Art. 118, equation 11)],

and consequently

$v=\frac{u_1}{u}=-F_1\frac{u^{\prime }}{u}$

is a solution of the equation in $v$.

The integral 10) of the last Chapter may be then written

${\delta }^{2}I=\underset{t_0}{\overset{t_1}{\int }}{F}_1{w}^2{(\frac{w^{\prime }}{w}-\frac{u^{\prime }}{u})}^2\text{d}t+{[R+w^2{F}_1\frac{u^{\prime }}{u}]}_{t_0}^{t_1}$

From this we see that if $\frac{w^{\prime }}{w}=\frac{u^{\prime }}{u}$, or if $w=Cu$, then the second variation is free from the sign of integration ; in other words, the second variation is free from the integral sign, if we make any deformation (normal [Art. 113, equation 5)] to the curve) such that the displacement is proportional to the value of any integral of the differential equation $J=0$.

Again, if we deform any one of the family of curves $G=0$ into a neighboring curve belonging to the family, we have an expression which is also free from the integral sign. For (see Arts. 79 and 81), if we write $p=\sqrt{x{\text{'}}^2+y{\text{'}}^2}=\frac{\text{d}s}{\text{d}t}$, we have

$\delta F=Gp{w}_N+{\left[\frac{\text{d}}{\text{d}t}(\xi \frac{\partial F}{\partial x^{\prime }}+\eta \frac{\partial F}{\partial y^{\prime }})\right]}_{t_0}^{t_1}$,

and consequently,

${\delta }^{2}F=p{w}_N\delta G+G\delta (p{w}_N)+{\left[\frac{\text{d}}{\text{d}t}\delta (\xi \frac{\partial F}{\partial x^{\prime }}+\eta \frac{\partial F}{\partial y^{\prime }})\right]}_{t_0}^{t_1}$.

Hence, if $\delta G=0$, we have here also

${\delta }^{2}I={\left[\delta (\xi \frac{\partial F}{\partial x^{\prime }}+\eta \frac{\partial F}{\partial y^{\prime }})\right]}_{t_0}^{t_1}$.

It may be shown as follows that the curve $\delta G=0$ is one of the family of curves $G=0$. The curves belonging to the family of curves $G=0$ are given (Art. 90) by

$x=\varphi (t,\alpha ,\beta ),y=\psi (t,\alpha ,\beta )$,

where $\alpha$ and $\beta$ are arbitrary constants. We have a neighboring curve of the family when for $\alpha$, $\beta$ we write $\alpha +ϵ{\alpha }^{\prime }$, $\beta +ϵ{\beta }^{\prime }$. Then the function $G$ becomes

$G+\mathrm{\Delta }G=G+ϵ\delta G+{ϵ}^2()+\dots$

Hence, when $ϵ$ is taken very small, it follows that

$x=\varphi (t,\alpha +ϵ{\alpha }^{\prime },\beta +ϵ{\beta }^{\prime }),y=\psi (t,\alpha +ϵ{\alpha }^{\prime },\beta +ϵ{\beta }^{\prime })$

is a solution of $\delta G=0$, since it is a solution of $G+\mathrm{\Delta }G=0$ and of $G=0$.

Now we may always choose normal displacements $\frac{w}{p}$ which will take us from one of the curves $G=0$ to a neighboring curve $\delta G=0$. From this it appears that there is a relation between the differential equations $\delta G=0$ and $J=0$.

Article 123.
In this connection a discovery made by Jacobi (Crelle's Journal, bd. 17, p. 68) is of great use. He showed that with the integration of the differential equation $G=0$, also that of the differential equation $J=0$ is performed. We are then able to derive the general expression for $u$, and may determine completely whether and when $u=0$. We shall next derive the general solution of the equation $J=0$, it being presupposed that the differential equation $G=0$ admits of a general solution. We derived the first variation in the form

$\delta I=\underset{t_0}{\overset{t_1}{\int }}Gw\text{d}t+[{]}_{t_0}^{t_1}$.

We may form the second variation by causing in this expression $G$ alone to vary, and then $w$ alone, and by adding the results.

It follows that

${\delta }^{2}I=\underset{t_0}{\overset{t_1}{\int }}(\delta Gw+G\delta w)\text{d}t+[{]}_{t_0}^{t_1}$. $$ (i)

Since the differential equation $G=0$ is supposed satisfied, we have

${\delta }^{2}I=\underset{t_0}{\overset{t_1}{\int }}\delta Gw\text{d}t+[{]}_{t_0}^{t_1}$. $$ (a)

We had (Art. 76)

$G_1=\frac{\partial F}{\partial x}-\frac{\text{d}}{\text{d}t}\left(\frac{\partial F}{\partial x^{\prime }}\right)$, $G_2=\frac{\partial F}{\partial y}-\frac{\text{d}}{\text{d}t}\left(\frac{\partial F}{\partial y^{\prime }}\right)$,

and also

$G_1=y^{\prime }G$, $G_2=-x^{\prime }G$.

When in the expression for $G_1$, the substitutions

$x\to x+ϵ\xi$, $y\to y+ϵ\eta$

are made, we have

$G_1+\mathrm{\Delta }{G}_1=(y^{\prime }+ϵ{\eta }^{\prime })(G+\mathrm{\Delta }G)$;

and since

$\mathrm{\Delta }{G}_1=ϵ\delta G_1+{ϵ}^2()+\cdots$,

$\mathrm{\Delta }G=ϵ\delta G+{ϵ}^2()+\cdots$,

it follows that

$\delta G_1=y^{\prime }\delta G=G{\eta }^{\prime }$,

and similarly

$\delta G_2=-x^{\prime }\delta G-G{\xi }^{\prime }$.

Article 124.
When $G$ is eliminated from the last two expressions, we have

$\delta G_1{\xi }^{\prime }+\delta G_2{\eta }^{\prime }=(y^{\prime }{\xi }^{\prime }-x^{\prime }{\eta }^{\prime })\delta G$. $(ii)$

On the other hand, it is seen that

$\delta G_1=\frac{{\partial }^{2}F}{\partial x^2}\xi +\frac{{\partial }^{2}F}{\partial x\partial y}\eta +\frac{{\partial }^{2}F}{\partial x\partial x^{\prime }}{\xi }^{\prime }+\frac{{\partial }^{2}F}{\partial x\partial y^{\prime }}{\eta }^{\prime }-\frac{\text{d}}{\text{d}t}(\frac{{\partial }^{2}F}{\partial x\partial x^{\prime }}\xi +\frac{{\partial }^{2}F}{{\partial }^2{x}^{\prime }}{\xi }^{\prime }+\frac{{\partial }^{2}F}{\partial x^{\prime }\partial y}\eta +\frac{{\partial }^{2}F}{\partial x^{\prime }\partial y}{\eta }^{\prime })$,

an expression which, owing to 2), 3) and 4) of the last Chapter, may be written in the following form :

$\delta G_1=\frac{{\partial }^{2}F}{\partial x^2}\xi +\frac{{\partial }^{2}F}{\partial x\partial y}\eta +\frac{{\partial }^{2}F}{\partial x\partial x^{\prime }}{\xi }^{\prime }+\frac{{\partial }^{2}F}{\partial x\partial y}{\eta }^{\prime }-\frac{\text{d}L}{\text{d}t}\xi -\frac{\text{d}M}{\text{d}t}\eta -L{\xi }^{\prime }-M{\eta }^{\prime }-\frac{\text{d}}{\text{d}t}\left(F_1{y}^{\prime }\frac{\text{d}w}{\text{d}t}\right)$;

and if we take into consideration 3), 4) 6) and 7) of the last Chapter, we may write the above result in the form:

$\delta G_1=-y^{\prime }\frac{\text{d}}{\text{d}t}\left(F_1\frac{\text{d}w}{\text{d}t}\right)+y^{\prime }{F}_{2}w$.

In an analogous manner, we have

$\delta G_2=x^{\prime }\frac{\text{d}}{\text{d}t}\left(F_1\frac{\text{d}w}{\text{d}t}\right)-x^{\prime }{F}_{2}w$.

When these values are substituted in $(ii)$, we have

$\delta G=-\frac{\text{d}}{\text{d}t}\left(F_1\frac{\text{d}w}{\text{d}t}\right)+F_{2}w$. $(b)$

Hence from (a) we have

${\delta }^{2}I=\underset{t_0}{\overset{t_1}{\int }}(-\frac{\text{d}}{\text{d}t}\left(F_1\frac{\text{d}w}{\text{d}t}\right)w+F_2{w}^2)\text{d}t+[{]}_{t_0}^{t_1}$.

By the previous method we found the second variation to be [see formula 8) of the last Chapter]

${\delta }^{2}I=\underset{t_0}{\overset{t_1}{\int }}(F_1{\left(\frac{\text{d}w}{\text{d}t}\right)}^2+F_2{w}^2)\text{d}t+[{]}_{t_0}^{t_1}$.

These two expressions should agree as to a constant term. The difference of the integrals is

$D=\underset{t_0}{\overset{t_1}{\int }}-\frac{\text{d}}{\text{d}t}\left(F_1\frac{\text{d}w}{\text{d}t}\right)w\text{d}t-\underset{t_0}{\overset{t_1}{\int }}{F}_2{\left(\frac{\text{d}w}{\text{d}t}\right)}^2\text{d}t$;

but since

$\int \frac{\text{d}}{\text{d}t}\left(F_1\frac{\text{d}w}{\text{d}t}\right)w\text{d}t=w{F}_1\frac{\text{d}w}{\text{d}t}-\int F_1{\left(\frac{\text{d}w}{\text{d}t}\right)}^2\text{d}t$,

it is seen that

$D={[-w{F}_1\frac{\text{d}w}{\text{d}t}]}_{t_0}^{t_1}$.

The formula (b) is

$\delta G=F_{2}w-\frac{\text{d}}{\text{d}t}\left(F_1\frac{\text{d}w}{\text{d}t}\right)$.

When we compare this with $12^a)$ of the preceding Chapter, the differential equation for <maht>u</math>, viz.:

$0=F_{2}u-\frac{\text{d}}{\text{d}t}\left(F_1\frac{\text{d}u}{\text{d}t}\right)$,

it is seen that as soon as we find a quantity $w$ for which $\delta G=0$, we have a corresponding integral of the diflEerential equation for $u$.

Article 125.
The total variation of $G$ is

$\mathrm{\Delta }G=G(x+ϵ{\xi }_1+\frac{{ϵ}^2}{2!}{\xi }_2+\cdots ,y+ϵ{\eta }_1+\frac{{ϵ}^2}{2!}{\eta }_2+\cdots ,x^{\prime }+ϵ\xi {\text{'}}_1+\frac{{ϵ}^2}{2!}\xi {\text{'}}_2+\cdots ,y^{\prime }+ϵ{\eta }_{1^{\prime }}+\frac{{ϵ}^2}{2!}{\eta }_{2^{\prime }}+\cdots ,x^{″}+ϵ{\xi }_{1^{\prime }}+\frac{{ϵ}^2}{2!}{\xi }^{\prime }{\text{'}}_2+\cdots ,y^{″}+ϵ{\eta }_{1^{″}}+\frac{{ϵ}^2}{2!}{\eta }_{2^{″}}+\cdots )-G(x,y,x^{\prime },y^{\prime },x^{″},y^{″})=ϵ\delta G=\frac{{ϵ}^2}{2!}{\delta }^{2}G+\cdots )$,

where $\delta G$, as found in the preceding article, has the value

$\delta G=-\frac{\text{d}}{\text{d}t}\left(F_1\frac{\text{d}w}{\text{d}t}\right)+F_{2}w$.

Suppose that the equation $G=0$ is integrable, and let

$x=\varphi (t,\alpha ,\beta )y=\psi (t,\alpha ,\beta )$

be general expressions which satisfy it, where $\alpha$, $\beta$ are arbitrary constants of integration. The difEerential equation $G=0$ will be satisfied, if we suppose that $\alpha$ and $\beta$, having arbitrarily fixed values, are increased by two arbitrarily small quantities $ϵ\delta \alpha$ and $ϵ\delta \beta$; that is, the functions

$\overline{x}=\varphi (t,\alpha +ϵ\delta \alpha ,\beta +ϵ\delta \beta )=\varphi (t,\alpha ,\beta )+ϵ(\frac{\partial \varphi }{\partial \alpha }\delta \alpha +\frac{\partial \varphi }{\partial \beta }\delta \beta )+{ϵ}^2()$,

$\overline{y}=\psi (t,\alpha +ϵ\delta \alpha ,\beta +ϵ\delta \beta )=\psi (t,\alpha ,\beta )+ϵ(\frac{\partial \psi }{\partial \alpha }\delta \alpha +\frac{\partial \psi }{\partial \beta }\delta \beta )+{ϵ}^2()$

are also solutions of $G=0$.

Article 126.
Now choose the variation of the curve (Art. Ill) in such a way that

$\overline{x}=x+ϵ{\xi }_1+\frac{{ϵ}^2}{2!}{\xi }_2+\cdots \overline{y}=y+ϵ{\eta }_1+\frac{{ϵ}^2}{2!}{\eta }_2+\cdots$;

and, whatever be the values of $\delta \alpha$ and $\delta \beta$, we determine ${\xi }_1$,${\xi }_2$,${\eta }_1$,${\eta }_2$, etc., by the relations:

${\xi }_1=\frac{\partial \varphi }{\partial \alpha }\delta \alpha +\frac{\partial \varphi }{\partial \beta }\delta \beta {\eta }_1=\frac{\partial \psi }{\partial \alpha }\delta \alpha +\frac{\partial \psi }{\partial \beta }\delta \beta$. $(iii)$

For all values of $\alpha$ and $\beta$ the difEerential equation $G=0$ satisfied; hence, the values of ${\xi }_1$, ${\eta }_1$, etc., just written, when substituted in $\mathrm{\Delta }G$ above must make the right-hand side of that equation vanish identically, and consequently also $\delta G$. Hence, the corresponding normal displacement $w=y^{\prime }{\xi }_1-x^{\prime }{\eta }_1$ transforms one of the system of curves $G=0$ to another one of the same system.

Since $\delta \alpha$ and $\delta \beta$ are entirely arbitrary, the coeflEcients of $\delta \alpha$ and $\delta \beta$ must each vanish in the expansion of $\mathrm{\Delta }G$ above. Owing to (iii) $w=y^{\prime }{\xi }_1-x^{\prime }{\eta }_1$ becomes

$w=(y^{\prime }\frac{\partial \varphi }{\partial \alpha }-x^{\prime }\frac{\partial \psi }{\partial \alpha })\delta \alpha +(y^{\prime }\frac{\partial \varphi }{\partial \beta }-x^{\prime }\frac{\partial \psi }{\partial \beta })\delta \beta$.

Writing this value of $w$ in the equation $\delta G=0$, we have

$-\frac{\text{d}}{\text{d}t}\left(F_1\frac{\text{d}}{\text{d}t}[(y^{\prime }\frac{\partial \varphi }{\partial \alpha }-x^{\prime }\frac{\partial \psi }{\partial \alpha })\delta \alpha +(y^{\prime }\frac{\partial \varphi }{\partial \beta }-x^{\prime }\frac{\partial \psi }{\partial \beta })\delta \beta ]\right)+F_2[(y^{\prime }\frac{\partial \varphi }{\partial \alpha }-x^{\prime }\frac{\partial \psi }{\partial \alpha })\delta \alpha +(y^{\prime }\frac{\partial \varphi }{\partial \beta }-x^{\prime }\frac{\partial \psi }{\partial \beta })\delta \beta ]=0$.

By equating the coefficients of $\delta \alpha$ and $\delta \beta$ respectively to zero, we have the two equations:

$1)-\frac{\text{d}}{\text{d}t}(F_1\frac{\text{d}}{\text{d}t}{\theta }_v(t))+F_2{\theta }_v(t)=0,(v=1,2)$

where, for brevity, we have written

$\frac{\partial \varphi (t)}{\partial t}={\varphi }^{\prime }(t),\frac{\partial \varphi }{\partial \alpha }={\varphi }_1(t),\frac{\partial \varphi }{\partial \beta }={\varphi }_2,\frac{\partial \psi (t)}{\partial t}={\psi }^{\prime }(t),\frac{\partial \psi }{\partial \alpha }={\psi }_1(t),\frac{\partial \psi }{\partial \beta }={\psi }_2$

$2){\theta }_1(t)={\varphi }^{\prime }(t){\varphi }_1(t)-{\varphi }^{\prime }(t){\psi }_1(t),{\theta }_2(t)={\psi }^{\prime }(t){\varphi }_2(t)-{\varphi }^{\prime }(t){\psi }_2(t)$.

It is seen at once that ${\theta }_1(t)$ and ${\theta }_2(t)$ are the solutions of the differential equation

$\frac{\text{d}}{\text{d}t}\left(F_1\frac{\text{d}u}{\text{d}t}\right)-F_{2}u=0$.

Hence it is seen that the general solution of the differential equation for $u$ is had from the integrals of the differential equation $G=0$, through simple differentiation.

Article 127.
We have next to prove that the two solutions ${\theta }_1(t)$ and ${\theta }_2(t)$ are independent of each other. In order to make this proof as simple as possible, let $x$ be written for the arbitrary quantity $t$.

Then the expressions $x=\varphi (t,\alpha ,\beta )$, $y=\psi (t,\alpha ,\beta )$, etc., become

$x=x,y=\psi (x,\alpha ,\beta ),$

${\varphi }^{\prime }=1,{\varphi }_1=0,{\varphi }_2=0,{\psi }^{\prime }=\frac{\text{d}y}{\text{d}x},$

${\theta }_1=-{\psi }_1,{\theta }_2=-{\psi }_2$.

If ${\theta }_1$ and ${\theta }_2$ are linearly dependent upon each other, we must have

${\theta }_2=\text{constant}{\theta }_1$,

from which it follows, at once, that

${\theta }_1{\theta }_{2^{\prime }}={\theta }_2{\theta }_{1^{\prime }}=0$,

where the accents denote differentiation with respect to $x$; or,

${\psi }_1{\psi }_{2^{\prime }}-{\psi }_2{\psi }_{1^{\prime }}=0$.

On the other hand, $y=\psi (x,\alpha ,\beta )$ is the complete solution of the differential equation, which arises out of $G_2=-x^{\prime }G=0$, when $x$ is written for $t$; that is, of

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

but here $\alpha$ and $\beta$ are two arbitrary independent constants, and consequently $\psi$ and ${\psi }^{\prime }=\frac{\text{d}\psi }{\text{d}x}$ are independent of each other with respect to $\alpha$ and $\beta$, so that the determinant

${\psi }_1{\psi }_{2^{\prime }}-{\psi }_2{\psi }_{1^{\prime }}$

is different from zero. Consequently ${\theta }_1$ and ${\theta }_2$ are independent of each other, since the contrary assumption stands in contradiction to the result just established. Hence, the general solution of the differential equation $J=0$, is of the form

$u=c_1{\theta }_1(t)+c_2{\theta }_2(t)$,

where $c_1$ and $c_2$ are arbitrary constants.

Article 128.
Following the methods of Weierstrass we have just proved the assertion of Jacobi ; since, as soon as we have the complete integral of $G=0$, it is easy to express the complete solution of the differential equation $J=0$.

The constants $c_1$ and $c_2$ may be so determined that $u$ vanishes on a definite position $t^{\prime }$, which may lie somewhere on the curve before we get to $t_1$. This may be effected by writing

$c_1=-{\theta }_2(t^{\prime }),c_2={\theta }_1(t^{\prime })$.

The solution of the equation $J=0$ becomes

$3)u={\theta }_1(t^{\prime }){\theta }_2(t)-{\theta }_2(t^{\prime }){\theta }_1(t)=\mathrm{\Theta }(t,t^{\prime })$.

It may turn out that $\mathrm{\Theta }(t,t^{\prime })$ vanishes for no other value of $t$; but it may also happen that there are other positions than $t^{\prime }$ at which $\mathrm{\Theta }(t,t^{\prime })$ becomes zero. If $t^{″}$ is the first zero position of $\mathrm{\Theta }(t,t^{\prime })$ which follows $t^{\prime }$ then $t^{″}$ is called the conjugate point to $t^{\prime }$.

Since $t^{\prime }$ has been arbitrarily chosen, we may associate with every point of the curve a second point, its conjugate. This being premised, we come to the following theorem, also due to Jacobi :

If within the interval $t_0\dots t_1$ there are no two points which are conjugate to each other in the above sense, then it is possible so to determine u that it satisfies the differential equation $J=0$, and nowhere vanishes within the interval $t_0\dots t_1$.

Article 129.
Let the point $t=t^{\prime }$ be a zero position of the function

$u=\mathrm{\Theta }(t,t^{\prime })$,

and let $t^{″}$ be a conjugate point to $t^{\prime }$, then $\mathrm{\Theta }(t,t^{\prime })$ will not again vanish within the interval $t^{\prime }\dots t^{″}$. Take in the neighborhood of the point $t^{\prime }$ a point $t^{\prime }+\tau$, where $\tau >0$, then the point which is conjugate to $t^{\prime }+\tau$ can lie only on the other side of $t^{″}$. This may be shown as follows:

If ${\displaystyle }$u = \Theta(t,t') is a solution of the equation

$F_1\frac{{\text{d}}^{2}u}{\text{d}{t}^2}+\frac{\text{d}{F}_1}{\text{d}t}\frac{\text{d}u}{\text{d}t}-F_{2}u=0$,

then is

$\overline{u}=\mathrm{\Theta }(t,t^{\prime }+\tau )$

a solution of the same equation ; that is, of

$F_1\frac{{\text{d}}^2\overline{u}}{\text{d}{t}^2}+\frac{\text{d}{F}_1}{\text{d}t}\frac{\text{d}\overline{u}}{\text{d}t}-F_2\overline{u}=0$,

since $\overline{u}$ differs from $u$ only through another choice of the arbitrary constants $c_1$ and $c_2$.

If $\tau$ is chosen sujBciently small, then $\mathrm{\Theta }(t^{\prime }+\tau ,t^{\prime })$ is different from zero and consequently also $\mathrm{\Theta }(t^{\prime },t^{\prime }+\tau )\ne 0$.

Eliminate $F_2$ from the two equations above, and we have

$4)F_1(u\frac{{\text{d}}^2\overline{u}}{\text{d}{t}^2}-\overline{u}\frac{{\text{d}}^{2}u}{\text{d}{t}^2})+\frac{\text{d}{F}_1}{\text{d}t}(u\frac{\text{d}\overline{u}}{\text{d}t}-\overline{u}\frac{\text{d}u}{\text{d}t})=0$.

Now write

$5)u\frac{\text{d}\overline{u}}{\text{d}t}-\overline{u}\frac{\text{d}u}{\text{d}t}=v$,

and the above equation becomes

$6)\frac{\text{d}v}{v}=-\frac{\text{d}{F}_1}{F_1}$,

which, when integrated, is

$7)v=u\frac{\text{d}\overline{u}}{\text{d}t}-\overline{u}\frac{\text{d}u}{\text{d}t}=+\frac{C}{F_1}$.

The constant $C$ in this expression cannot vanish, for, in that case,

$u=\text{const.}\overline{u}$,

or

$\mathrm{\Theta }(t,t^{\prime })=\text{const.}\mathrm{\Theta }(t,t^{\prime }+\tau )$.

Since, however, $\mathrm{\Theta }(t,t^{\prime })$ vanishes for $t=t^{\prime }$, it results from the above that $\mathrm{\Theta }(t^{\prime },t^{\prime }+\tau )=0$, which is contrary to the hypothesis, and consequently $C$ cannot vanish.

It is further assumed that $F_1$ does not change its sign or become zero within the interval $t_0\dots t_1$. If $F_1$ vanishes without a transition from the positive to the negative or vice versa within the stretch $t_0\dots t_1$ then in general no further deductions can be drawn, and a special investigation has to be made for each particular case.

In the first case, however, $v$ has a finite value, and the equation 7), when divided through by $u^2$ becomes

$\frac{u\frac{\text{d}\overline{u}}{\text{d}t}-\overline{u}\frac{\text{d}u}{\text{d}t}}{u^2}=\frac{\text{d}\frac{\overline{u}}{u}}{\text{d}t}=\frac{C}{F_1{u}^2}$,

an expression, which, when integrated, is

$\overline{u}=Cu\underset{t^{\prime }+\tau }{\int }\frac{\text{d}t}{F_1{u}^2}$.

Since the function $u$ does not vanish between $t^{\prime }$ and $t^{″}$, it follows from the last expression that $\overline{u}$ cannot vanish between the limits $t^{\prime }+\tau$ and $t^{″}$. Accordingly, if there is a point conjugate to $t^{\prime }+\tau$, it cannot lie before $t^{″}$. If, therefore, we choose a point $t^{‴}$ before $t^{″}$ and as close to it as we wish, then $u^{\prime }$ will certainly not vanish within the interval $t^{\prime }+\tau \dots t^{‴}$.

If $t^{\prime }$ is a point situated immediately before $t_0$, and if we determine the point $t^{″}$ conjugate to $t^{\prime }$, and choose a point $t_1$ before $t^{″}$ and as near to it as we wish, then from the preceding it is clear that no points conjugate to each other lie within the interval $t_0\dots t_1$, the boundaries excluded. We may then, as shown above, find a function $u$, which satisfies the differential equation $J=0$ and which vanishes neither on the limits nor within the interval $t_0\dots t_1$. The transformation of Art. 117 is therefore admissible, and the sign of ${\delta }^{2}I$ depends only upon the sign of $F_1$.

Article 130.
We may investigate a little more closely the relation of Art. 120, where

$u_2\frac{\text{d}{u}_1}{\text{d}t}-u_1\frac{\text{d}{u}_2}{\text{d}t}=\frac{C}{f_1}$.

In the interval under consideration, boundaries included, we assume that $F_1$ does not become zero or infinite, and consequently retains the same sign. Further, the constant $C$ has always the same value and is different from zero, since $u_1$ and $u_2$ are linearly independent.

It follows at once that $\frac{\text{d}{u}_1}{\text{d}t}$ cannot be zero at the same time that $u_1$ is zero; for then $C$ would be zero contrary to our hypothesis.

Owing to the form

$\frac{\text{d}}{\text{d}t}\left(\frac{u_1}{u_2}\right)=\frac{1}{u_2^2}\frac{C}{F_1}$,

it is clear that $\frac{\text{d}}{\text{d}t}\left(\frac{u_1}{u_2}\right)$ has the same sign as $\frac{C}{F_1}$. We may take this sign positive, since otherwise owing to the expression

$u_1\frac{\text{d}{u}_2}{\text{d}t}-u_2\frac{\text{d}{u}_1}{\text{d}t}=\frac{C}{F_1}$

we would would have $\frac{\text{d}}{\text{d}t}\left(\frac{u_2}{u_1}\right)$ positive. We may assume then that the indices have been placed upon the $u$'s, so that $\frac{u_1}{u_2}$ is always on the increase with increasing t.

The ratio $\frac{u_1}{u_2}$ will become infinite for the zero values of $u_2$ (see Art. 120). Since this quotient is always increasing with increasing values of $t$, the trace of the corresponding curve must pass through $+\mathrm{\infty }$, and return again (if it does return) from $-\mathrm{\infty }$. Values of $t$, for which this quotient has the same value, may be called congruent.

It is evident, as shown in the accompanying figure, that such values are equi-distant from two values of $t$, say $t_0$ and $t_1$, which make $u_2=0$. The abscissae are values of $t$, and the ordinates are the corresponding values of the ratio $\frac{u_1}{u_2}$.

Article 131.
To summarize : We have supposed the cases excluded in which $F_1$ is zero along the curve under consideration. If this function were zero at an isolated point of the curve, it would be a limiting case of what we have considered. If it were zero along a stretch of this curve, we should have to consider variations of the third order, and would have, in general, neither a maximum nor a minimum value unless this variation also vanished, leaving us to investigate variations of the fourth order. We exclude these cases from the present treatment, and suppose also that $F_1$ and $F_2$ are everywhere finite along our curve (otherwise the expression for the second variation, viz. ”

$\int (F_{1}w{\text{'}}^2+F_2{w}^2)\text{d}t$,

would have no meaning).

We also derived in Art. 124 the variation of $G$ in the form

$\delta G=F_{2}w-\frac{\text{d}}{\text{d}t}\left(F_1\frac{\text{d}w}{\text{d}t}\right)$,

and when this is compared with the differential equation

$12^a)0=F_{2}u-\frac{\text{d}}{\text{d}t}\left(F_1\frac{\text{d}u}{\text{d}t}\right)$ (see Art. 118),

it is seen that if an integral $u$ of the differential equation $12^a)$ vanishes for any value of $t$, the corresponding integral $w$ of the equation $\delta G=0$ vanishes for the same value of $t$.

In Art. 126 we had

$w=y^{\prime }{\xi }_1-x^{\prime }{\eta }_1=\delta \alpha {\theta }_1(t)+\delta \beta {\theta }_2(t)$,

where the displacement ${\xi }_1$, ${\eta }_1$ takes us from a point of the curve $G=0$ to a point of the curve $\delta G=0$. Consequently the normal displacement $w_N$ can be zero only at a point where the curves $G=0$ and $\delta G=0$ intersect.

At such a point we must have

$\delta \alpha {\theta }_1(t)\delta \beta {\theta }_2(t)=0$.

When one of the family of curves $G=0$ has been selected, the two associated constants $\alpha$ and $\beta$ are fixed. These are the constants that occur in ${\theta }_1(t)$ and ${\theta }_2(t)$. If , further, the curve passes through a fixed point $P_0$, the variable $t$ is determined, and consequently the functions ${\theta }_1(t)$ and ${\theta }_2(t)$ are definitely determined, so that the ratio $\delta \alpha :\delta \beta$ is definitely known from the above relation. There may be a second point at which the curves $G=0$ and $\delta G=0$ intersect. This point is the point conjugate to $P_0$ (see Art. 128).

Article 132.
The geometrical significance of these conjugate points is more fully considered in Chapter XI. Writing the second variation in the form

${\delta }^{2}I=\underset{t_0}{\overset{t_1}{\int }}{F}_1{w}^2(\frac{w^{\prime }}{w}-\frac{u^{\prime }}{u})\text{d}t$,

we see that the possibility of $\frac{w^{\prime }}{w}-\frac{u^{\prime }}{u}=0$ is when $u=Cw$. Now $w$ is zero at both of the end-points of the curve, since at these points there is no variation, but $u$ is equal to zero at $P_1$ only when $P_1$ is conjugate to $P_0$. Hence, unless the two curves $G=0$ and ${\displaystyle }$\delta G = 0 intersect again at $P_1$, $u$ is not equal to zero at $P_1$, and consequently

${(\frac{w^{\prime }}{w}-\frac{u^{\prime }}{u})}^2\ne 0$.

In this case, if $F_1$ has a positive sign throughout the interval $t_0\dots t_1$, there is a possibility of a minimum value of the integral $I$, and there is a possibility of a maximum value when $F_1$ has a negative sign throughout this interval.

Article 133.
Next, let $P_1$ be conjugate to $P_0$, so that at both of the limits of integration we have $u=0=w$. We may then take $u=w$ at all other points of the curve, so that consequently

${\delta }^{2}I=\underset{t_0}{\overset{t_1}{\int }}{F}_1{w}^2{(\frac{w^{\prime }}{w}-\frac{u^{\prime }}{u})}^2\text{d}t=0$.

We cannot then say anything regarding a maximum or a minimum until we have investigated the variations of a higher order.

Next, suppose that a pair of conjugate points are situated between $P_0$ and $P_1$, and let these points be $P^{\prime }$ and $p^{″}$. We may then make a displacement of the curve so that

$w=kw$ from $P_0$ to $P^{\prime }$,

$w=u+kw$ from $P^{\prime }$ to $P^{″}$ and

$w=kw$ from $P^{″}$ to $P_1$,

where $k$ is an indeterminate constant. The quantity $w$ is subjected only to the condition that it must be zero at $P_0$ and $P_1$, and $u$ must be a solution of the difEerential equation $J=0$, and is zero at the conjugate points $P^{\prime }$ and $P^{″}$.

The second variation takes the form

${\delta }^{2}I=k^2\underset{t_0}{\overset{t^{\prime }}{\int }}(F_{1}w{\text{'}}^2+F_2{w}^2)\text{d}t+\underset{t^{\prime }}{\overset{t^{″}}{\int }}[(F_{1}u{\text{'}}^2+F_2{u}^2)+2k(F_1{u}^{\prime }{w}^{\prime }+F_{2}uw)+k^2(F_{1}w{\text{'}}^2+F_2{w}^2)]\text{d}t+k\underset{t^{″}}{\overset{t_1}{\int }}(F_{1}w{\text{'}}^2+F_2{w}^2)\text{d}t$.

In the preceding article we saw (cf. also Art. 117) that

$\underset{t^{\prime }}{\overset{t^{″}}{\int }}(F_{1}u{\text{'}}^2+F_2{u}^2)\text{d}t=0$,

and we may therefore write ${\delta }^{2}I$ in the form

${\delta }^{2}I=2k\int (F_1{u}^{\prime }{w}^{\prime }+F_{2}uw)\text{d}t+k^{2}M$,

where $M$ is a finite quantity.

The integral

$\underset{t^{\prime }}{\overset{t^{″}}{\int }}(F_1{u}^{\prime }{w}^{\prime }+F_{2}uw)\text{d}t$

may be written

$\underset{t^{\prime }}{\overset{t^{″}}{\int }}(-\frac{\text{d}}{\text{d}t}(F_1{u}^{\prime })+F_{2}u)w\text{d}t+[F_1{u}^{\prime }w{]}_{t^{\prime }}^{t^{″}}$

and since, in virtue of the formula $12^a)$ of Art. 118, the expression under this latter integral sign is zero, it follows that

${\delta }^{2}I=2k[F_1{u}^{\prime }w{]}_{t^{\prime }}^{t^{″}}+k^{2}M$.

Further, by hypothesis, $F_1$ retains the same sign within the interval $t^{\prime }\dots t^{″}$, and does not become zero within or at these limits, the function $u^{\prime }$ is different from zero at the limits (Arts. 130 and 152), and of opposite sign at these limits, since $u$, always retaining the same sign, leaves the value zero at one limit and approaches it at the other limit. Consequently $[F_1{u}^{\prime }]$ is finite and of opposite signs at the two points $P^{\prime }$ and $P^{″}$, and it remains only that $w$ be chosen finite and with the same sign, so that $[F_1{u}^{\prime }w{]}_{t^{\prime }}^{t^{″}}$ be different from zero. Hence by the proper choice of $k$ we may effect displacements for which ${\delta }^{2}I$ is positive, and also those for which it is negative.

Hence when our interval includes not, however, both as extremities) a pair of conjugate points, we have definitely established that the curve in question can give rise to neither a maximum nor a minimum.

The above semi-geometrical proof is due to a note given by Prof. Schwarz at Berlin (1898-99); see also Lefon V of a course of Lectures given by Prof.Picard at Paris (1899-1900) on "Equations aux dirivies partielles."