CHAPTER XVIII: PROOF OF TWO THEOREMS WHICH HAVE BEEN ASSUMED IN THE PREVIOUS CHAPTER.
- 230 The two theorems proved in the present Chapter are:
- 1) That possible to construct a portion of space about a curve which satisfies the differential equation of the problem in such a way that it is always possible to join any point in this limited space and the initial-point by one and only one curve which likewise satisfies the differential equation.
- 2) That the function cannot vanish along an entire curve within such a portion of space.
- 231 The equations of the curve in space. The coordinates expressed in power-series.
- 232 The curve determined through its initial-point and initial direction. Condition that two curves pass through the same initial-point and have initial directions differing by a given small quantity.
- 233 Condition that one of these curves pass through a point which is in the neighborhood of a point on the other curve. The determinant which arises from these conditions.
- 234 Conjugate points in connection with this determinant.
- 235 Case where the end-points are conjugate to each other.
- 236 The differential equation and the determinant .
- 237 Simplification of the determinant .
- 238,239,240,241,242 The determinant changes sign when it vanishes.
- 241 An exceptional case.
- 243 The point conjugate to the initial-point is the limit of the intersection of neighboring curves that pass through the initial-point.
- 244 When in the determinant , the quantity varies in a continuous manner, the point conjugate to varies in a continuous manner.
- 245,246,247,248 Proof of the theorem that a portion of curve including conjugate points may be so varied that the total variation of one of the integrals may be either positive or negative, while the other integral remains unchanged.
- 249,250,251,252 The function cannot vanish along an entire curve within the portion of space defined above.
- 253,254,255,256 The -function for the case of the Isoperimetrical Problem.
- 256,257 The -function for the curve whose center of gravity lies lowest.
In the present Chapter proofs are given of the theorems:
. That it is possible to construct a portion of space about a curve, which satisfies the differential equation of the problem in such a way that it is always possible to join any point in this limited space and the initial point by one and only one curve which likewise satisfies the differential equation.
. The function cannot vanish along an entire curve within such a portion of space.
Let the coordinates of a curve which satisfies the differential equation be expressed as functions of a quality . These functions contain three arbitrary constants: the two constants and of integration and the constant . If then and are the coordinates of the corresponding point in space, we have
where corresponds to the point . By changing the three constants we have another curve in space. The requirement that the projection of this latter curve should go through the point gives two relations between the increments of the constants where is the value of in the new equation that corresponds to the point .
The equations of the new curve in space are :
and, if are the coordinates of ,
These equations represent for sufficiently small values of , which satisfy the last two equations, all curves in space which satisfy the differential equations and whose projection upon the -plane in its initial direction deviates very little from the initial direction of the projection of the original curve.
We may express and as power-series in and the trigonometrical tangent of the angle which the two initial directions form with each other. If this tangent is denoted by , we have, as in Art. 148,
Since the two curves are to go through the same initial point, we have further
The determinant of the linear terms on the right-hand side of the equations 4) and 5) is
In this determinant write
If we multiply the second horizontal row by , the third by and add both to the first, the determinant may then be written
This quantity is not zero, as we shall see later [see the third of of equations 13) in Art. 237].
Hence we may express as power-series in so that for any pair of values , which have been taken sufficiently small, there corresponds a curve in space.
From the differential equation it follows in a similar manner as was shown in Art. 149, that for one pair of values there corresponds only one curve, and that every curve is completely determined through the initial point and the initial direction. We, therefore, conclude, as in Art. 149, that the equations 4) and 5) afford us all the curves which are neighboring the original curve, which have the same end-point with it, and which satisfy the differential equation.
We have now to choose the constants in such a way that the new curve in space will go through a point which lies in the neighborhood of any point situated on the old curve. If then we give to / a definite value and take sufficiently small values for , the following equations must be satisfied:
Hence, if we write
it follows that
If we substitute instead of and their power-series in in equation 9), the determinant of the linear terms on the righthand side of equations 8) and 9) become after a slight transformation
We assume that this determinant does not vanish for arbitraryvalues of . This case and the formulae which would follow from it we leave as an exception for future investigation.
The first value of after for which vanishes we call the conjugate to .
We see then that if the upper limit of the integrals lies before the point that is conjugate to . the curve can envelop a portion of space having the property desired.
Since in this case, if are chosen sufficiently small, one can always express as power-series in and consequently can construct one and only one curve in space which satisfies the differential equation, which passes through the point and the point and which deviates in its position arbitrarily little from the original curve. To these difEerent curves in space there correspond different functions . If, however, the curves lie sufficiently near the original curve, the functions which correspond to them will not vanish for any point along them, so that through any point in a sufficiently small neighborhood of any point of these curves a curve starting from can be drawn which satisfies the differential equation.
It remains yet to be proved that, if the point conjugate to lies between and , we cannot have a maximum or a minimum value of the integral.
Since the point can be chosen arbitrarily near 0 and since the point conjugate to varies in a continuous manner with it is necessary only to show that cannot lie between 0 and the point conjugate to it. We then will have proved everything except the case where coincides with the point conjugate to 0. This case we must again leave for a special investigation, since the curve may or may not offer a maximum or a minimum, (cf. Art. 132.)
A rigorous proof of what has been said requires a close investigation of the function .
The curve in space which we had through variation of the constants is determined through the initial direction of its projection at the point and through the differential equation , which it must satisfy. From this the properties of the function may also be inferred.
We perform the changes which suffers when undergo the changes . The equation must vanish for arbitrary values of .
In a similar manner as was shown on page 133, formula (b), we have
The terms of the first dimension in the development of in powers of are
which we represent by . We then have
Since this quantity must be zero for arbitrary values of , the coefficients of the individual terms in this expression when developed in power-series must be zero. If we limit ourselves to the linear terms, and use the functional sign for the function itself when there can be no confusion, we have the following three differential equations:
If we multiply the first of these equations by , the second by and add the results, we have
Similarly, if we multiply the second equation by and the third by we have upon adding,
Finally, if we multiply the first equation by and the second by , we have through addition,
From these equations it follows that
The constant cannot be zero ; for then we would have
But, as is easily shown, the determinant may be brought to the form
If then , it would also follow that , and the determinant would vanish, since two vertical rows differ from each other only by a constant factor; and this is true for arbitrary values of , which case we have excluded. Hence the constant cannot be zero.
We next prove that the determinant changes sign, when it vanishes. We have
Owing to 8), we have
Consequently the first of the determinants vanishes, leaving
We introduce the following notation:
We can then write
The numerator of the right-hand side of the above expression is equal to multiplied by the square of a certain expression, which we shall now determine.
Let us write
From 16) it follows at once that
Similarly, we have
Accordingly, the expression 18) may be written:
But owing to the relations 13)
it follows that
Further we have
But owing to the third relation in 13) the expression is independent of , so that the first term of the right-hand expression is zero, and consequently
Hence the equation 22) becomes
Suppose that is zero of the 'th order for the value so that the development of begins with to the 'th power.
If then does not vanish for , the development of
begins with the 'th power. But this expression is equal to , and since according to our assumptions does not become zero or infinity for any point within the interval , it is seen that must begin with an even power. Hence is an even integer, and consequently is an odd integer, and therefore must change signs when it vanishes.
Suppose next that vanishes for , then cannot vanish; for from the equations [see 16)]
it would follow, if and are not simultaneously zero, that
But this equation, as also the simultaneous vanishing of and for the value , contradicts the equation 13)
for, as we have seen, is difiFerent from 0 and is neither zero nor infinity.
Hence and do not vanish simultaneously. If then vanishes for , the development of in powers of begins with the first power.
We may therefore write
It follows that the development of
begins with the term
except when , in which case the coeflBcient of this term is zero. In this case nothing has been shown, but see the next article.
For it is evident that changes sign on vanishing.
We shall next show that if vanishes, can be zero only when at the same time . We saw in the preceding article that the quantities and cannot both be zero. If then , it follows, from the relation [formula 21)]
that, when and also , and also from
Similarly, when , it is seen from the equation
that , if , and, consequently, also .
But if does not vanish for , then , and, consequently, also does not vanish for . We have thus shown that D(io i) does not vanish for . It follows, therefore, that D(to, /) changes sign on vanishing except when we have simultaneously
In this case it has not been proved whether it changes sign or does not. We must, consequently, consider each separate case for itself (see Art. 255).
If we assume that at least one of the quantities is different from zero, we can give the geometrical significance of conjugate points :
When the constants are increased by , new curves in space are produced. The condition that one of these curves cuts the original curve in the point is [see equations 7) and 9)] expressed through the following equations :
or, if we eliminate and ,
The elimination of and gives
If is different from zero, we may take for a limit as small as we wish such that, for every whose absolute value is less than the prescribed limit, we always have
Consequently no value of can be found which satisfies equation 26).
If then is a definite value of for which is different from zero, there will be no value of within a certain interval which satisfies the equation 26). Hence among all the curves in space for which have sufficiently small values there will be none which cuts the original curve in the neighborhood of .
It is quite different, however, if we take for an interval which contains , the point conjugate to , within which, therefore, . For then has opposite signs for and . Hence after an arbitrarily small value has been fixed, we can always choose so small that also
has opposite signs for and , and consequently there will be within this interval a value of for which the equation
Hence, if we limit an interval ever so small about the point conjugate to and take arbitrarily small upper limits for , then among the admissible curves there are always such which start from and cut the original curve within this interval. Indeed, if are less than a certain quantity, then all the curves in space, for which have values not greater than this fixed quantity and which go through the point , cut the original curve within this interval. This upper limit for becomes infinitely small at the same time with this interval, so that the point conjugate to can be defined as the point which the points of intersection of neighboring curves approach.
In a similar manner we may prove that a portion of space as small as we choose may be taken around a point of the curve in space which is not conjugate to , and that the points along the curves in space, which are conjugate to , do not lie within this limited portion of space, if are taken sufficiently small; but when we limit a portion of space as small as we wish about the point that is conjugate to , the points along the curves in space that are conjugate to will with sufficiently small all lie within this interval.
It also follows that, if in the quantity varies in a continuous manner, the first value of , for which vanishes, varies in a continuous manner. This follows at once from
where becomes infinitely small with for every value of .
For and , where is the point conjugate to the point , the function has different signs, however small is; and if we take t sufficiently small, it follows that has different signs for and and must therefore vanish for some value of within the interval . The change in the conjugate point is consequently arbitrarily small for a sufficiently small increment in .
We come next to the proof of the theorem that a portion of curve which includes and the point conjugate to it may always be so varied that may be both positive and negative, while remains unchanged.
Let us write as in Arts. 180, 181:
We have accordingly
Now choose so that
Then from the condition that , it follows that we may express as a power-series in which begins with a power higher than the first in .
Hence (see Art. 180), it follows that
and, from Art. 189, that
or, what is the same thing :
where ; is an arbitrarily small quantity over which we have yet a choice.
We shall now show that if lies beyond the point conjugate to , the absolute value of may be chosen so small that besides satisfying the condition
the quantity will satisfy also the condition
without being everywhere zero. If is chosen sufficiently small, which we are always able to do, it is then seen that the quantity has the same sign as , and consequently the same sign as which may be either positive or negative.
Since vanishes for and , and since
it follows that instead of 29), we may write:
and instead of this equation and the equation
we may write the two equations:
where is a quantity independent of .
Now let be three functions of t which satisfy the three differential equations :
It follows from the theory of differential equations that for a series of values of for which is neither zero nor infinite differ from the three functions by quantities which become infinitely small at the same time with k.
Again, let be the point conjugate to . and write for the stretch from to , where is a point situated before the point ,
and for the stretch from to let
It is clear that is not everywhere zero unless , since owing to the differential equations 31) which satisfy, a linear relation for the series of values of can exist only if for these values , a case which we excluded (Art. 180).
The quantity satisfies the differential equation
It must also satisfy the additional conditions that for and for , and that
But we have
If we write
then from what was seen above, the functions differ from by a quantity which becomes infinitely small with .
The conditions which remain to be fulfilled are
The determinant of these equations differs from by a quantity which becomes infinitely small with (Art. 237).
For and the quantity has different signs, and consequently we may take so small that the determinant of the equation 32) has different signs for and and consequently vanishes for a value of situated between and .
We may therefore take along the curve before in such a way that the equations 32) are satisfied by values of and , which are not all zero.
If then, returning to equation 28), is chosen sufficiently small, it follows that has the sign of and since this is arbitrary, there are among the admissible variations of the curves those for which has a negative increm-ent and also those for which the increment of is positive.
The portion of curve 01 cannot therefore extend beyond the point which is conjugate to 0. If we exclude the case where 1 coincides exactly with the point that is conjugate to 0 it follows that 1 must lie before the point that is conjugate to 0. We may then choose so near to 0 that 1 lies also before the point that is conjtigate to . Along such a portion of curve the ftmction does not vanish and consequently we may envelop such a portion of curve in a portion of space which has the required properties.
It only remains, excluding exceptional cases, to show that the function cannot vanish along an entire curve within the portion of space defined above.
If we exclude the possibility of the integral
becoming zero along a portion of the curve in question, then for to vanish, it is necessary that along the whole curve; that is, the direction of the projection of the arbitrary curve in space must coincide at every point with the direction of the projection of the curve that satisfies the differential equation.
If are the coordinates of the two curves expressed as functions of their lengths of arc and , say, then at the point in question we must have
it follow,s also that
i. e., the two curves in space have at every point also the same direction.
represent the coordinates of the neighboring curve in space in the neighborhood of the point .
The point is now taken as any arbitrary point. If in the above expressions we consider as functions of a quantity which become infinitely small with , then for successive values of these expressions are the coordinates of the points of a certain curve which goes through ; and indeed every curve that passes through can be expressed in this manner, if the functions of are suitably chosen.
If now there is to be a curve in space along which , then its direction at the point , as we saw above, must coincide at this point with the direction of the curve which satisfies the differential equation determined through .
The direction-cosines of the latter curve are proportional to the following quantities:
and those of the first curve to:
where in ony and are to be considered as dependent on .
where in are to be considered as dependent upon , the increment due to being already explicitly expressed. If we integrate by parts the expression that stands under the integral sign of the last of the above quantities, and take into consideration the definitions 8) of Art. 233, it is seen that the direction-cosines of the arbitrary curve are proportional to the quantities
If the direction of the two curves are to coincide at the point in question, then the three minors formed from the quantities and must vanish. But these minors are identical with the minors formed from the quantities
Accordingly, the three quantities of the first row are proportional to the corresponding quantities of the second row.
If we make , the above quantities become
Hence, if we let denote the factor of proportionality, we have
where the third equation is reduced to this form b) the application of the other two.
Since the curve which satisfies the differential equation must pass through the point 4. we must in virtue of equations 5) and 6) have the relation
and from this it follows that
Eliminate from the first two equations in 33) and write for the differences that appear their values in terms of the 's defined by the relations 6).
The determinant of the resulting equation, of the last of the equations 33), and of equation 34) is identical with [See formula 14), Art. 237]. Hence if does not vanish, these equations have no other solution except
The same conclusions may also be drawn from any small value of to which the values correspond; there enters here instead of the quantity the quantity
Since this determinant for sufficiently small values of is different from zero, it also follows that
But the quantities do not vary with and are consequently zero, since they are zero for ; this means that the curve along which the function is to be zero must coincide with the original curve which satisfies the differential equation. It is then no variation of this curve and the arbitrariness of the quantities which is essential to the meaning of the function is entirely lost.
If , then would be the point conjugate to ; and since this would be true for every point of the arbitrary curve, it would follow that the arbitrary curve was formed from the points that are the conjugates of the initial point and would consequently lie without or at least on the boundary of the portion of space under consideration.
It is seen that there is no curve within our assumed portion of space along which the function vanishes everywhere. The purport of this Chapter is thus completed. It has also been shown that the conditions necessary and sufficient for the existence of a maximum or a minimum are in the Theory of Relative Maxima and Minima the analogues of those enumerated in Art. 174.
We shall now finish the proof of the maximal and minimal properties of the two problems already considered, viz., the isoperimetrical problem and the problem of finding the curve whose center of gravity lies lowest.
In the case of the isoperimetrical problem we have
In these expressions is a positive constant; further, and do not vanish simultaneously at any point of the curve, so there is no exceptional case which requires a special investigation. The expression is the cosine of the angle between the two directions and , so that has for no point and no direction a positive value. As we have already seen, this is one of the requirements for a maximum.
Let two points 0 and 1 be connected by the arc of a circle of given length, which together with a fixed curve, that joins the two points, incloses a surface area. The integral taken over the whole periphery is represented by (see Art. 191)
and it is required to prove that one cannot connect the two points by a curve of the same length which includes a greater area with the fixed curve. The proof is immediate as soon as the following is shown. If an arbitrary curve of the prescribed length is drawn between the two points, then we may draw through any point 2 of this curve the arc of a circle which also goes through 0 and which has the same length as the portion of the arbitrary curve situated between 0 and 2
If we let the point 2 traverse the arbitrary curve, the successive arcs of circles are variations of one another and their lengths differ indefinitely little from one another. If the point 2 is sufficiently near the initial point, the corresponding arc of circle becomes the arc of circle for which the maximal property is to be proved.
This is all done as soon as we stipulate that each arc of circle is to be traversed once only and is to be constructed as indicated in Art. 229. Then indeed there is only one arc of circle having the given length that can be laid between the points 0 and 2. The arcs of circles corresponding to the successive lengths are variations of one another, and their lengths at corresponding points differ indefinitely little from one another, and consequently the arcs of circles, if the point 2 coincides with , pass in a continuous manner into the original arc of circle drawn between 0 and 1.
Regarding the determinant we have here
From these expressions we have
It is seen from this that the first time after that vanishes, is for the value ; consequently is the point conjugate to the initial-point.
In reality, if we consider the initial and the end-point coinciding so that the curve satisfying the differential equation is a complete circle, then this curve does no longer offer a maximum, at least, in the sense that with every arbitrarily small variation of the curve the variation would be smaller; since we could slide at pleasure the curve congruent to itself, and therefore vary the curve without altering the perimeter or the surface area.
It is interesting to observe that a case appears in this problem which could not be decided in the general treatment; namely, where simultaneously vanish with (see Art. 242).
In reality for , we have
Nevertheless, changes sign when it passes through zero; for the vanishing of is effected by making the factor zero. But this factor changes sign, while the second factor retains its sign for .
In the problem of finding the curve whose center of gravity lies lowest, we had (Art. 216)
We saw that , and further and do not vanish simultaneously at any point Consequently is everywhere different from 0 and . Since represents the cosine of the angle between the two directions and , its absolute value cannot exceed unity, and in general is less than unity, so that the function is nowhere negative, as must be the case for a minimum. We have already seen in Art. 219, if the length of arc is sufficiently great, that between two arbitrarily given points one curve and only one may be drawn which satisfies the differential equation. It then follows that there can be no conjugate points and consequently the catenary in its whole trace has the desired minimal property.
That there are no conjugate points is also seen from the consideration of the determinant . For we have
From these quantities we have D(ta,t=”J” Pi 7T multiplied by the determinant
- multiplied by the determinant
The equation , or
has no real root except , that is, there exists no point conjugate to the point .