Calculus of Variations/CHAPTER XVII

CHAPTER XVII: THE SUFFICIENT CONDITIONS.

• 22 The problem solved without using the second variation.
• 221 The ${\mathcal {E}}$-function.
• 222 Consequences due to this function.
• 223 A field about the curve which maximizes or minimizes the integral.
• 224 Further discussion of the nature of the enveloped space.
• 225 Properties of an enveloped space which lies within the first enveloped space.
• 226 The sufficiency of the ${\mathcal {E}}$ condition.
• 227 It is assumed that through the initial-point and any other point in the enveloped space only one curve may be drawn, which satisfies the differential equation (cf. Article 230).
• 228 The ${\mathcal {E}}$-function cannot be zero along an entire curve that lies within the enveloped space (cf. Article 230).
• 229 Extension of the meaning- of the integrals that have been employed.

Article 220.
In a similar manner as in the case of free variation (see Art. 159) there is also a way of solving completely the general problem of restricted variation without making use of the second variation. Let the differential equation be found through the variation of the integrals. The required curve must necessarily satisfy this equation. Let the portion of curve under consideration be so limited that for every point of it $F^{(0)}$ and $F^{(1)}$ are regular functions in $x,y,x',y'$ and for no point on it the function $F_{1}$ becomes zero or infinite. The case where the portion of curve contains singular points will be left for a special investigation in each particular problem.

Article 221.
Let and 1 be the end-points of the portion of curve in question. Through an arbitrary point 2 of this curve we draw any regular curve and on it take a point 3 so near to 2 that we may join 0 and 3 by a curve which satisfies the differential equation. The line 0 3 2 1 is a possible variation of 0 1. The change which an integral

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

suffers through this variation, takes the form (Art. 205)

$\Delta I^{(0)}={\mathcal {E}}^{(0)}(x_{2},y_{2},p_{2},q_{2},{\bar {p}}_{2},{\bar {q}}_{2})\sigma +\int _{t_{0}}^{t_{1}}G^{(0)}w{\text{d}}t+\left(\sigma ,\xi ,\eta ,{\frac {{\text{d}}\xi }{{\text{d}}t}},{\frac {{\text{d}}\eta }{{\text{d}}t}}\right)_{2}$

where $\sigma$ is the length of 2 3 taken in the positive direction; $p_{2},q_{2}$ denote the direction-cosines of 02 at 2; ${\bar {p}}_{2},{\bar {q}}_{2}$, those of $3'2$ at 2 and $x_{2},y_{2}$ the coordinates of 2.

If we have two integrals, and if the variation is such that one of the two integrals remains unchanged, and if $I_{(1)}$, ${\mathcal {E}}^{(1)}$, $G^{(1)}$ denote the corresponding quantities for the second integral, then we have

$\Delta I^{(0)}={\mathcal {E}}^{(0)}\sigma +\int _{t_{0}}^{t_{1}}G^{(0)}w{\text{d}}t+\left(\sigma ,\xi ,\eta ,{\frac {{\text{d}}\xi }{{\text{d}}t}},{\frac {{\text{d}}\eta }{{\text{d}}t}}\right)_{2}$

$0={\mathcal {E}}^{(1)}\sigma +\int _{t_{0}}^{t_{1}}G^{(1)}w{\text{d}}t+\left(\sigma ,\xi ,\eta ,{\frac {{\text{d}}\xi }{{\text{d}}t}},{\frac {{\text{d}}\eta }{{\text{d}}t}}\right)_{2}$

Hence if, as usual, we denote the quantity ${\mathcal {(0)}}-\lambda {\mathcal {E}}^{(1)}$ by ${\mathcal {E}}$, then is

$\Delta I^{(0)}={\mathcal {E}}\sigma +\int _{t_{0}}^{t_{1}}(G^{(0)}-\lambda G^{(1)}){\text{d}}t+\left(\sigma ,\xi ,\eta ,{\frac {{\text{d}}\xi }{{\text{d}}t}},{\frac {{\text{d}}\eta }{{\text{d}}t}}\right)_{2}$

and, if the curve satisfies the differential equation $G^{(0)}-\lambda G^{(1)}=0$, it follows that

$\Delta I^{(0)}={\mathcal {E}}\sigma +\left(\sigma ,\xi ,\eta ,{\frac {{\text{d}}\xi }{{\text{d}}t}},{\frac {{\text{d}}\eta }{{\text{d}}t}}\right)_{2}$

From this equation it is seen that the function ${\mathcal {E}}(x,y,p,q,{\bar {p}},{\bar {q}})$ along the whole portion of curve cannot have opposite signs for any two pairs of values ${\bar {p}},{\bar {q}}$, as $\Delta I^{(0)}$ must always have the same sign.

Article 222.
As in Art. 157, we may write ${\mathcal {E}}$ in the form

${\mathcal {E}}(x,t,p,q,{\bar {p}},{\bar {q}})=(q{\bar {p}}-p{\bar {q}})^{2}\int _{0}^{1}[F_{1}^{(0)}(x,y,p_{k},q_{k})-\lambda F_{1}^{(1)}(x,y,p_{k},q_{k})](1-k){\text{d}}k$

where $p_{k}=(1-k)p+k{\bar {p}}\qquad q_{k}=(1-k)q+k{\bar {q}}$

It follows at once from the preceding Article that

$F_{1}^{(0)}(x,y,p_{k},q_{k})-\lambda F_{1}^{(1)}(x,y,p_{k},q_{k})$

cannot have values with diflEerent signs for any values of $p,q$.

The converse, however, is not true. (See Art. 160). The condition that ${\mathcal {E}}$ cannot change its sign in so far as every arbitary direction ${\bar {p}},{\bar {q}}$ is concerned has a further significance.

For erect lines along the curve 01 perpendicular to the plane of this curve. On these perpendiculars take lengths equal in value to the second integral, where in each case the integration is taken from 0 to the foot of the perpendicular. Then to the curve 01 there corresponds a curve in space $01'$, where the points in space are marked by indices corresponding to the points in the plane.

Thus to every curve through the point 0 and lying in this plane there corresponds a curve in space. We say that a curve in space satisfies the difEerential equation of the problem if its projection satisfies the differential equation $G^{(0)}-\lambda G^{(1)}=0$, although $\lambda$ need not have the same value for all the curves.

Article 223.
Now suppose that we can envelop the curve $01'$ in space in the following manner : The point 0 is to lie on the boundary, and the point 1' within the space enveloped ; further, it is to be possible to draw from 0 to every point within this enveloped space at least one curve which satisfies the differential equation ; and, when such a curve has been drawn from 0 to any point $P$ within the enveloped space, it must be possible to draw a curve between 0 and a point neighboring to $P$ which also satisfies the differential equation. This curve must lie everywhere as near as we wish to the first curve, and the associated $\lambda$'s can differ from one another only by arbitrarily small quantities.

If the end-point describes a continuous curve in the enveloped space, then we may draw a series of curves, corresponding to the successive positions of the end-point, which satisfy the differential equation.

Article 224.
We shall show in the next Chapter that there must exist an enveloped space as described above, if the curve 01 is to offer a maximum or a minimum. There are exceptional cases which are to be treated separately.

We may at first assume the existence of such a space in order to make the essential points as clear as possible. We saw above that the function ${\mathcal {E}}(x,y,p,q,{\bar {p}},{\bar {q}})$ along the whole curve for arbitrary values of ${\bar {p}},{\bar {q}}$ could not have different signs. From this we infer that in general ${\mathcal {E}}(x,y,p,q,{\bar {p}},{\bar {q}})$ will not have values with different signs for other curves which satisf the differential equation. The deviation in the directions of these curves from the position of the original curve, of course, lies within certain limits, and the corresponding $\lambda$'s vary suflciently little from the $\lambda$ of the original curve. This will certainly be true if the integral

$\int _{0}^{1}[F_{1}^{(0)}(x,y,p_{k},q_{k})-\lambda F_{1}^{(1)}(x,y,p_{k},q_{k})](1-k){\text{d}}k$

is everywhere different from zero along the first curve.

Excepting the case where the above integral becomes zero, we have as a further necessary condition that it must be possible to envelop the portion of curve 01 by a portion of surface, on the boundary of which the point 0 lies, so that within this portion of surface the function ${\mathcal {E}}(x,y,p,q,{\bar {p}},{\bar {q}})$ does not have values with different signs along any of the curves that pass through 0, and lie within the portion of surface in question, it being assumed that they all satisfy the differential equation, and that the difference in value of $\lambda$ is suflciently small for all the curves. (See Art. 156).

Article 225.
The same considerations are also true for a point ${\bar {0}}$ which lies before 0 along the same curve 01, so that then 01 lies wholly within the corresponding portion of surface, and our original enveloped space, including the point 1', may be so formed as to lie wholly within the space enveloped by this second surface.

Suppose that the integration of the above integrals begins now with the point ${\bar {0}}$ instead of with the point 0 as before. Keeping our former notation, let the point 0' correspond in space to 0 and join 0' and 1' by a regular curve which lies wholly within the enveloped space.

This curve is quite arbitrary and is subjected to the condition that if 2 is the projection of any point 2' upon the $xy$-plane, the sum

$I_{{\bar {0}}0}^{(1)}+{\bar {I}}_{02}^{(1)}$

is equal to the length of the perpendicular projecting the point 2', where we use the notation $I$ to represent an integral that is taken over a definite curve that satisfies the differential equation and ${\bar {I}}$ one that is taken over an arbitrary curve, and where the indices represent the limits and the direction of the integration. A curve that satisfies the differential equation may be drawn from ${\bar {0}}$ to every point 2' of the curve in the enveloped space and this curve also with the exception of the point ${\bar {0}}$ lies within the enveloped portion of space. These curves are to have the property, which after the assumptions is always possible, that beginning with ${\bar {0}}1'$ the following curves are always variations of the preceding.

Article 226.
We regard the coordinates of the points of the projection of the arbitrary curve as functions of the length of arc counted from the point 1, and we consider the sum $I_{{\bar {0}}2}+{\bar {I}}_{21}$.

It follows from the fixed relation regarding the point in space that, wherever the point 2 may lie upon the curve 021, we always have

$I_{{\bar {0}}2}^{(1)}+{\bar {I}}_{{\bar {2}}1}^{(1)}=I_{{\bar {0}}01}^{(1)}$

There is consequently no variation in the integral $I^{(1)}$.

Let the length of the portion 1 2 increase by $\sigma$. The change thereby produced in $I_{{\bar {0}}2}+{\bar {I}}_{21}$ is equal to

${\mathcal {E}}(x_{2},y_{2},p_{2},q_{2},{\bar {p}}_{2},{\bar {q}}_{2})+\left(\sigma ,\xi ,\eta ,{\frac {{\text{d}}\xi }{{\text{d}}t}},{\frac {{\text{d}}\eta }{{\text{d}}t}}\right)_{2}$

where $p_{2},q_{2}$ are the direction-cosines of ${\bar {0}}2$ at $2,{\bar {p}}_{2},{\bar {q}}_{2}$ those of 1 2 at 2. Again let the length of arc 1 2 decrease by $\sigma$. The change thereby experienced in $I_{{\bar {0}}2}+{\bar {I}}_{21}$ is equal to

$-{\mathcal {E}}(x_{2},y_{2},p_{2},q_{2},{\bar {p}}_{2},{\bar {q}}_{2})+\left(\sigma ,\xi ,\eta ,{\frac {{\text{d}}\xi }{{\text{d}}t}},{\frac {{\text{d}}\eta }{{\text{d}}t}}\right)_{2}$

Since $\xi ,\eta ,{\frac {{\text{d}}\xi }{{\text{d}}t}},{\frac {{\text{d}}\eta }{{\text{d}}t}}$ become indefinitely small with $\sigma$, the quantity ${\mathcal {E}}(x_{2},y_{2},p_{2},q_{2},{\bar {p}}_{2},{\bar {q}}_{2})$ represents the differential quotient of the sum $I_{{\bar {0}}2}+{\bar {I}}_{21}$, this sum being considered as a function of the length of arc 12 (see Art. 161).

If the point 2 coincides with 1, then is $I_{{\bar {0}}2}+{\bar {I}}_{21}=I_{01}$, and if 2 coincides with 0, we have

$I_{{\bar {0}}2}+{\bar {I}}_{21}=I_{{\bar {0}}0}+{\bar {I}}_{01}$

Hence it follows :

1) If ${\mathcal {E}}$ along $021$ is not positive and not everywhere zero, that

${\displaystyle }$

2) If ${\mathcal {E}}$ along $021$ is not negative and not everywhere zero, that

${\displaystyle }$

Article 227.
It will be shown in the next Chapter that we may assume the strips of surface enveloping ${\bar {0}}01$ so narrow that the point ${\bar {0}}$ may be joined with any other point within this enveloped space by one curve, and only one, which satisfies the differential equation. The curve must of course lie wholly within the enveloped space. This assumed, it follows that the integral $I_{01}$ is identical with $I_{{\bar {0}}01}$, and further, that the integral $I_{{\bar {0}}0}$ is identical with the portion of the integral $I_{{\bar {0}}01}$, which is taken over the portion of curve ${\bar {0}}0$.

We therefore have

in case 1) $I_{01}>{\bar {I}}_{01}$;

in case 2) $I_{01}<{\bar {I}}_{01}$.

The maximal and minimal property of the curve that satisfies the differential equation is accordingly proved except in the case where along the whole curve the function ${\mathcal {E}}$ is zero.

Article 228.
We shall show that for the case where in the constructed realm the integral

$\int _{0}^{1}(F_{1}^{(0)}-\lambda F_{1}^{(1)})(1-k){\text{d}}k$

is not zero, the function ${\mathcal {E}}$ cannot be zero along a whole curve between 0 and 1; or what is the same, that it is not possible everywhere along the curve to have

$p{\bar {q}}-{\bar {p}}q=0$

After we have proved this theorem for a regular curve, we may extend it for a curve composed of regular portions; since, as has often been shown, sudden changes in the direction along the curve under consideration have no influence upon the deductions that have been drawn.

Finally, the same is also true for arbitrary curves which can be drawn between 0 and 1 and which lie suflBciently near the curve that satisfies the differential equation, but this is true in so far only as the two integrals have a meaning for these curves.

Article 229.
In a similar manner as was shown in Art. 165, the meaning of the integrals may be extended, if for these integrals are substituted sums of integrals which are taken over portions of regular curves that join a series of points on the curves under consideration. It remains then to show, when these sums approach finite fixed values by increasing the number of points and diminishing the distance between such points, that these limiting values are at the same time the values of the original integrals. Of course, the limiting value which is thus determined for the second integral $I^{(1)}$ must be identical with the value that was prescribed for it.

If the integrals taken over an arbitrary curve have in this sense a definite value, it is clear that, for example in case of a maximum, the integral $I^{(0)}$ taken over this curve cannot be greater than the integral taken over the curve which satisfies the differential equation. For we could form a curve out of regular portions of curve, the integral over which would be as little different from the integral $I^{(0)}$ as we wished, and consequently would also be greater than the integral taken over the curve which satisfies the differential equation. This is not possible after the hypothesis. Hence, that integral must be smaller than this one, as we may again show as follows : Iet 2 be a point of the arbitrary curve sufficiently near 01, then we may draw two portions of curve which satisfy the differential equation, the one from 0 to 2 and the other from 2 to 1, the corresponding curves in space being 0' 2' and 2' 1'. Around 0' 2' and 2' 1' we may limit a portion of space in a similar manner as was done around 0' 1' and with the analogous properties. If the portion of space about 0' 1' is taken sufficiently small, the arbitrary curve will lie within the portion of space which envelops 0' 2' and2'l'.

The integral taken over this curve is, consequently after what was given above, not greater than the integral taken over the two portions of curve which satisfy the differential equation ; but this is smaller than $I_{01}$, and consequently also the integral of the arbitary curve is smaller than $I_{01}$.

We must point out here a limitation which has been tacitly made : We traced the curve 0 2 1 in such a way that the corresponding curve in space lay in the portion of space defined above. Now, there may be curves 02 1 in a region about 01 taken arbitrarily small, such that the corresponding curves in space do not fall within the limited portion of space, and for such variations the maximal and minimal properties are not derived through our conclusion». Our proof has reference only to such variations in which the change of the value of the second integral becomes indefinitely small at corresponding points at the same time with the variation of the coordinates.