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 u, 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 $$$$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 $$$$\delta G = 0 intersect again at $$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."