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 $\Theta (t,t')$. 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 $t_{0}\ldots t_{1}$, suppose that two functions $\phi _{1}(t)$ and $\phi _{2}(t)$ can be found which satisfy the differential equation 13) of the last Chapter, so that, consequently,

$u=c_{1}\phi _{1}(t)+c_{2}\phi _{2}(t)$

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 $t_{0}\ldots t_{1}$.

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

$u_{1}=-F_{1}u'\qquad$ [see Art. 118, equation 11)],

and consequently

$v={\frac {u_{1}}{u}}=-F_{1}{\frac {u'}{u}}$

is a solution of the equation in $v$.

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

$\delta ^{2}I=\int _{t_{0}}^{t_{1}}F_{1}w^{2}\left({\frac {w'}{w}}-{\frac {u'}{u}}\right)^{2}~{\text{d}}t+\left[R+w^{2}F_{1}{\frac {u'}{u}}\right]_{t_{0}}^{t_{1}}$

From this we see that if ${\frac {w'}{w}}={\frac {u'}{u}}$, 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 $p={\sqrt {x'^{2}+y'^{2}}}={\frac {{\text{d}}s}{{\text{d}}t}}$, we have

$\delta F=Gpw_{N}+\left[{\frac {\text{d}}{{\text{d}}t}}\left(\xi {\frac {\partial F}{\partial x'}}+\eta {\frac {\partial F}{\partial y'}}\right)\right]_{t_{0}}^{t_{1}}$,

and consequently,

$\delta ^{2}F=pw_{N}\delta G+G\delta (pw_{N})+\left[{\frac {\text{d}}{{\text{d}}t}}\delta \left(\xi {\frac {\partial F}{\partial x'}}+\eta {\frac {\partial F}{\partial y'}}\right)\right]_{t_{0}}^{t_{1}}$.

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

$\delta ^{2}I=\left[\delta \left(\xi {\frac {\partial F}{\partial x'}}+\eta {\frac {\partial F}{\partial y'}}\right)\right]_{t_{0}}^{t_{1}}$.

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

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

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

$G+\Delta G=G+\epsilon \delta G+\epsilon ^{2}(~~)+\ldots$

Hence, when $\epsilon$ is taken very small, it follows that

$x=\phi (t,\alpha +\epsilon \alpha ',\beta +\epsilon \beta '),\qquad y=\psi (t,\alpha +\epsilon \alpha ',\beta +\epsilon \beta ')$

is a solution of $\delta G=0$, since it is a solution of $G+\Delta G=0$ 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

$\delta I=\int _{t_{0}}^{t_{1}}Gw~{\text{d}}t+{\Big [}~~{\Big ]}_{t_{0}}^{t_{1}}$.

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

$\delta ^{2}I=\int _{t_{0}}^{t_{1}}(\delta Gw+G\delta w)~{\text{d}}t+{\Big [}~~{\Big ]}_{t_{0}}^{t_{1}}$. $\qquad$ (i)

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

$\delta ^{2}I=\int _{t_{0}}^{t_{1}}\delta Gw~{\text{d}}t+{\Big [}~~{\Big ]}_{t_{0}}^{t_{1}}$. $\qquad$ (a)

We had (Art. 76)

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

and also

$G_{1}=y'G$, $\qquad G_{2}=-x'G$.

When in the expression for $G_{1}$, the substitutions

$x\rightarrow x+\epsilon \xi$, $\qquad y\rightarrow y+\epsilon \eta$

are made, we have

$G_{1}+\Delta G_{1}=(y'+\epsilon \eta ')(G+\Delta G)$;

and since

$\Delta G_{1}=\epsilon \delta G_{1}+\epsilon ^{2}(~~)+\cdots$,

$\Delta G=\epsilon \delta G+\epsilon ^{2}(~~)+\cdots$,

it follows that

$\delta G_{1}=y'\delta G=G\eta '$,

and similarly

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

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

$\delta G_{1}\xi '+\delta G_{2}\eta '=(y'\xi '-x'\eta ')\delta G$. $\qquad (ii)$

On the other hand, it is seen that

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

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

$\delta G_{1}={\frac {\partial ^{2}F}{\partial x^{2}}}\xi +{\frac {\partial ^{2}F}{\partial x\partial y}}\eta +{\frac {\partial ^{2}F}{\partial x\partial x'}}\xi '+{\frac {\partial ^{2}F}{\partial x\partial y}}\eta '-{\frac {{\text{d}}L}{{\text{d}}t}}\xi -{\frac {{\text{d}}M}{{\text{d}}t}}\eta -L\xi '-M\eta '-{\frac {\text{d}}{{\text{d}}t}}\left(F_{1}y'{\frac {{\text{d}}w}{{\text{d}}t}}\right)$;

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

$\delta G_{1}=-y'{\frac {\text{d}}{{\text{d}}t}}\left(F_{1}{\frac {{\text{d}}w}{{\text{d}}t}}\right)+y'F_{2}w$.

In an analogous manner, we have

$\delta G_{2}=x'{\frac {\text{d}}{{\text{d}}t}}\left(F_{1}{\frac {{\text{d}}w}{{\text{d}}t}}\right)-x'F_{2}w$.

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

$\delta G=-{\frac {\text{d}}{{\text{d}}t}}\left(F_{1}{\frac {{\text{d}}w}{{\text{d}}t}}\right)+F_{2}w$. $\qquad (b)$

Hence from (a) we have

$\delta ^{2}I=\int _{t_{0}}^{t_{1}}\left(-{\frac {\text{d}}{{\text{d}}t}}\left(F_{1}{\frac {{\text{d}}w}{{\text{d}}t}}\right)w+F_{2}w^{2}\right)~{\text{d}}t+{\Big [}~~{\Big ]}_{t_{0}}^{t_{1}}$.

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

$\delta ^{2}I=\int _{t_{0}}^{t_{1}}\left(F_{1}\left({\frac {{\text{d}}w}{{\text{d}}t}}\right)^{2}+F_{2}w^{2}\right)~{\text{d}}t+{\Big [}~~{\Big ]}_{t_{0}}^{t_{1}}$.

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

$D=\int _{t_{0}}^{t_{1}}-{\frac {\text{d}}{{\text{d}}t}}\left(F_{1}{\frac {{\text{d}}w}{{\text{d}}t}}\right)w~{\text{d}}t-\int _{t_{0}}^{t_{1}}F_{2}\left({\frac {{\text{d}}w}{{\text{d}}t}}\right)^{2}~{\text{d}}t$;

but since

$\int {\frac {\text{d}}{{\text{d}}t}}\left(F_{1}{\frac {{\text{d}}w}{{\text{d}}t}}\right)w~{\text{d}}t=wF_{1}{\frac {{\text{d}}w}{{\text{d}}t}}-\int F_{1}\left({\frac {{\text{d}}w}{{\text{d}}t}}\right)^{2}~{\text{d}}t$,

it is seen that

$D=\left[-wF_{1}{\frac {{\text{d}}w}{{\text{d}}t}}\right]_{t_{0}}^{t_{1}}$.

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

$\Delta G=G\left(x+\epsilon \xi _{1}+{\frac {\epsilon ^{2}}{2!}}\xi _{2}+\cdots ,y+\epsilon \eta _{1}+{\frac {\epsilon ^{2}}{2!}}\eta _{2}+\cdots ,x'+\epsilon \xi '_{1}+{\frac {\epsilon ^{2}}{2!}}\xi '_{2}+\cdots ,y'+\epsilon \eta _{1}'+{\frac {\epsilon ^{2}}{2!}}\eta _{2}'+\cdots ,x''+\epsilon \xi _{1}'+{\frac {\epsilon ^{2}}{2!}}\xi ''_{2}+\cdots ,y''+\epsilon \eta _{1}''+{\frac {\epsilon ^{2}}{2!}}\eta _{2}''+\cdots \right)-G(x,y,x',y',x'',y'')=\epsilon \delta G={\frac {\epsilon ^{2}}{2!}}\delta ^{2}G+\cdots )$,

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

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

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 $\epsilon \delta \alpha$ and $\epsilon \delta \beta$; that is, the functions

${\bar {x}}=\phi (t,\alpha +\epsilon \delta \alpha ,\beta +\epsilon \delta \beta )=\phi (t,\alpha ,\beta )+\epsilon \left({\frac {\partial \phi }{\partial \alpha }}\delta \alpha +{\frac {\partial \phi }{\partial \beta }}\delta \beta \right)+\epsilon ^{2}(~~)$,

${\bar {y}}=\psi (t,\alpha +\epsilon \delta \alpha ,\beta +\epsilon \delta \beta )=\psi (t,\alpha ,\beta )+\epsilon \left({\frac {\partial \psi }{\partial \alpha }}\delta \alpha +{\frac {\partial \psi }{\partial \beta }}\delta \beta \right)+\epsilon ^{2}(~~)$

are also solutions of $G=0$.

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

${\bar {x}}=x+\epsilon \xi _{1}+{\frac {\epsilon ^{2}}{2!}}\xi _{2}+\cdots \qquad {\bar {y}}=y+\epsilon \eta _{1}+{\frac {\epsilon ^{2}}{2!}}\eta _{2}+\cdots$;

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:

$\xi _{1}={\frac {\partial \phi }{\partial \alpha }}\delta \alpha +{\frac {\partial \phi }{\partial \beta }}\delta \beta \qquad \eta _{1}={\frac {\partial \psi }{\partial \alpha }}\delta \alpha +{\frac {\partial \psi }{\partial \beta }}\delta \beta$. $\qquad (iii)$

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 $\Delta G$ above must make the right-hand side of that equation vanish identically, and consequently also $\delta G$. Hence, the corresponding normal displacement $w=y'\xi _{1}-x'\eta _{1}$ 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 $\Delta G$ above. Owing to (iii) $w=y'\xi _{1}-x'\eta _{1}$ becomes

$w=\left(y'{\frac {\partial \phi }{\partial \alpha }}-x'{\frac {\partial \psi }{\partial \alpha }}\right)\delta \alpha +\left(y'{\frac {\partial \phi }{\partial \beta }}-x'{\frac {\partial \psi }{\partial \beta }}\right)\delta \beta$.

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

$-{\frac {\text{d}}{{\text{d}}t}}\left(F_{1}{\frac {\text{d}}{{\text{d}}t}}\left[\left(y'{\frac {\partial \phi }{\partial \alpha }}-x'{\frac {\partial \psi }{\partial \alpha }}\right)\delta \alpha +\left(y'{\frac {\partial \phi }{\partial \beta }}-x'{\frac {\partial \psi }{\partial \beta }}\right)\delta \beta \right]\right)+F_{2}\left[\left(y'{\frac {\partial \phi }{\partial \alpha }}-x'{\frac {\partial \psi }{\partial \alpha }}\right)\delta \alpha +\left(y'{\frac {\partial \phi }{\partial \beta }}-x'{\frac {\partial \psi }{\partial \beta }}\right)\delta \beta \right]=0$.

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

$1)\qquad -{\frac {\text{d}}{{\text{d}}t}}\left(F_{1}{\frac {\text{d}}{{\text{d}}t}}\theta _{v}(t)\right)+F_{2}\theta _{v}(t)=0,\qquad (v=1,2)$

where, for brevity, we have written

${\frac {\partial \phi (t)}{\partial t}}=\phi '(t),{\frac {\partial \phi }{\partial \alpha }}=\phi _{1}(t),{\frac {\partial \phi }{\partial \beta }}=\phi _{2},\qquad {\frac {\partial \psi (t)}{\partial t}}=\psi '(t),{\frac {\partial \psi }{\partial \alpha }}=\psi _{1}(t),{\frac {\partial \psi }{\partial \beta }}=\psi _{2}$

$2)\qquad \theta _{1}(t)=\phi '(t)\phi _{1}(t)-\phi '(t)\psi _{1}(t),\qquad \theta _{2}(t)=\psi '(t)\phi _{2}(t)-\phi '(t)\psi _{2}(t)$.

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 $x=\phi (t,\alpha ,\beta )$, $y=\psi (t,\alpha ,\beta )$, etc., become

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

$\phi '=1,\qquad \phi _{1}=0,\phi _{2}=0,\qquad \psi '={\frac {{\text{d}}y}{{\text{d}}x}},$

$\theta _{1}=-\psi _{1},\theta _{2}=-\psi _{2}$.

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

$\theta _{1}\theta _{2}'=\theta _{2}\theta _{1}'=0$,

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

$\psi _{1}\psi _{2}'-\psi _{2}\psi _{1}'=0$.

On the other hand, $y=\psi (x,\alpha ,\beta )$ is the complete solution of the differential equation, which arises out of $G_{2}=-x'G=0$, 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 $\psi '={\frac {{\text{d}}\psi }{{\text{d}}x}}$ are independent of each other with respect to $\alpha$ and $\beta$, so that the determinant

$\psi _{1}\psi _{2}'-\psi _{2}\psi _{1}'$

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 $t'$, which may lie somewhere on the curve before we get to $t_{1}$. This may be effected by writing

$c_{1}=-\theta _{2}(t'),\qquad c_{2}=\theta _{1}(t')$.

The solution of the equation $J=0$ becomes

$3)\qquad u=\theta _{1}(t')\theta _{2}(t)-\theta _{2}(t')\theta _{1}(t)=\Theta (t,t')$.

It may turn out that $\Theta (t,t')$ vanishes for no other value of $t$; but it may also happen that there are other positions than $t'$ at which $\Theta (t,t')$ becomes zero. If $t''$ is the first zero position of $\Theta (t,t')$ which follows $t'$ then $t''$ is called the conjugate point to $t'$.

Since $t'$ 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 $t_{0}\ldots t_{1}$ 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 $t_{0}\ldots t_{1}$.

Article 129.
Let the point $t=t'$ be a zero position of the function

$u=\Theta (t,t')$,

and let $t''$ be a conjugate point to $t'$, then $\Theta (t,t')$ will not again vanish within the interval $t'\ldots t''$. Take in the neighborhood of the point $t'$ a point $t'+\tau$, where $\tau >0$, then the point which is conjugate to $t'+\tau$ 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

${\bar {u}}=\Theta (t,t'+\tau )$

a solution of the same equation ; that is, of

$F_{1}{\frac {{\text{d}}^{2}{\bar {u}}}{{\text{d}}t^{2}}}+{\frac {{\text{d}}F_{1}}{{\text{d}}t}}{\frac {{\text{d}}{\bar {u}}}{{\text{d}}t}}-F_{2}{\bar {u}}=0$,

since ${\bar {u}}$ differs from $u$ only through another choice of the arbitrary constants $c_{1}$ and $c_{2}$.

If $\tau$ is chosen sujBciently small, then $\Theta (t'+\tau ,t')$ is different from zero and consequently also $\Theta (t',t'+\tau )\neq 0$.

Eliminate $F_{2}$ from the two equations above, and we have

$4)\qquad F_{1}\left(u{\frac {{\text{d}}^{2}{\bar {u}}}{{\text{d}}t^{2}}}-{\bar {u}}{\frac {{\text{d}}^{2}u}{{\text{d}}t^{2}}}\right)+{\frac {{\text{d}}F_{1}}{{\text{d}}t}}\left(u{\frac {{\text{d}}{\bar {u}}}{{\text{d}}t}}-{\bar {u}}{\frac {{\text{d}}u}{{\text{d}}t}}\right)=0$.

Now write

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

and the above equation becomes

$6){\frac {{\text{d}}v}{v}}=-{\frac {{\text{d}}F_{1}}{F_{1}}}$,

which, when integrated, is

$7)\qquad v=u{\frac {{\text{d}}{\bar {u}}}{{\text{d}}t}}-{\bar {u}}{\frac {{\text{d}}u}{{\text{d}}t}}=+{\frac {C}{F_{1}}}$.

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

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

or

$\Theta (t,t')={\text{const.}}\Theta (t,t'+\tau )$.

Since, however, $\Theta (t,t')$ vanishes for $t=t'$, it results from the above that $\Theta (t',t'+\tau )=0$, 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 $t_{0}\ldots t_{1}$. If $F_{1}$ vanishes without a transition from the positive to the negative or vice versa within the stretch $t_{0}\ldots t_{1}$ 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

${\frac {u{\frac {{\text{d}}{\bar {u}}}{{\text{d}}t}}-{\bar {u}}{\frac {{\text{d}}u}{{\text{d}}t}}}{u^{2}}}={\frac {{\text{d}}{\frac {\bar {u}}{u}}}{{\text{d}}t}}={\frac {C}{F_{1}u^{2}}}$,

an expression, which, when integrated, is

${\bar {u}}=Cu\int _{t'+\tau }{\frac {{\text{d}}t}{F_{1}u^{2}}}$.

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

If $t'$ is a point situated immediately before $t_{0}$, and if we determine the point $t''$ conjugate to $t'$, 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 $t_{0}\ldots t_{1}$, 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 $t_{0}\ldots t_{1}$. 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 $+\infty$, and return again (if it does return) from $-\infty$. 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. ”

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

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

$12^{a})\qquad 0=F_{2}u-{\frac {\text{d}}{{\text{d}}t}}\left(F_{1}{\frac {{\text{d}}u}{{\text{d}}t}}\right)\qquad$ (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

$w=y'\xi _{1}-x'\eta _{1}=\delta \alpha \theta _{1}(t)+\delta \beta \theta _{2}(t)$,

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

$\delta ^{2}I=\int _{t_{0}}^{t_{1}}F_{1}w^{2}\left({\frac {w'}{w}}-{\frac {u'}{u}}\right)~{\text{d}}t$,

we see that the possibility of ${\frac {w'}{w}}-{\frac {u'}{u}}=0$ 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

$\left({\frac {w'}{w}}-{\frac {u'}{u}}\right)^{2}\neq 0$.

In this case, if $F_{1}$ has a positive sign throughout the interval $t_{0}\ldots t_{1}$, 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

$\delta ^{2}I=\int _{t_{0}}^{t_{1}}F_{1}w^{2}\left({\frac {w'}{w}}-{\frac {u'}{u}}\right)^{2}~{\text{d}}t=0$.

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 $P'$ and $p''$. We may then make a displacement of the curve so that

$w=kw$ from $P_{0}$ to $P'$,

$w=u+kw$ from $P'$ 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 $P'$ and $P''$.

The second variation takes the form

$\delta ^{2}I=k^{2}\int _{t_{0}}^{t'}(F_{1}w'^{2}+F_{2}w^{2})~{\text{d}}t+\int _{t'}^{t''}[(F_{1}u'^{2}+F_{2}u^{2})+2k(F_{1}u'w'+F_{2}uw)+k^{2}(F_{1}w'^{2}+F_{2}w^{2})]~{\text{d}}t+k\int _{t''}^{t_{1}}(F_{1}w'^{2}+F_{2}w^{2})~{\text{d}}t$.

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

$\int _{t'}^{t''}(F_{1}u'^{2}+F_{2}u^{2})~{\text{d}}t=0$,

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

$\delta ^{2}I=2k\int (F_{1}u'w'+F_{2}uw)~{\text{d}}t+k^{2}M$,

where $M$ is a finite quantity.

The integral

$\int _{t'}^{t''}(F_{1}u'w'+F_{2}uw)~{\text{d}}t$

may be written

$\int _{t'}^{t''}\left(-{\frac {\text{d}}{{\text{d}}t}}(F_{1}u')+F_{2}u\right)w~{\text{d}}t+{\Big [}F_{1}u'w{\Big ]}_{t'}^{t''}$

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

$\delta ^{2}I=2k{\Big [}F_{1}u'w{\Big ]}_{t'}^{t''}+k^{2}M$.

Further, by hypothesis, $F_{1}$ retains the same sign within the interval $t'\ldots t''$, and does not become zero within or at these limits, the function $u'$ 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 $[F_{1}u']$ is finite and of opposite signs at the two points $P'$ and $P''$, and it remains only that $w$ be chosen finite and with the same sign, so that ${\Big [}F_{1}u'w{\Big ]}_{t'}^{t''}$ 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."