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.
The condition given in the preceding Chapter is not sufficient to establish the existence of a maximum or a minimum. Under the assumption that is neither zero nor infinite within the interval , suppose that two functions and can be found which satisfy the differential equation 13) of the last Chapter, so that, consequently,
is the general solution of . Then, even if within the limits of integration it can be shown that is not infinite, it may still happen that, however the constants and be chosen, the function vanishes, so that the transformation of the -equation into the -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 defined by the equation , in order that we may determine under what conditions this function does not vanish within the interval .
It is seen that the equation is satisfied, if for we write
[see Art. 118, equation 11)],
is a solution of the equation in .
The integral 10) of the last Chapter may be then written
From this we see that if , or if , 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 .
Again, if we deform any one of the family of curves 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 , we have
Hence, if , we have here also
It may be shown as follows that the curve is one of the family of curves . The curves belonging to the family of curves are given (Art. 90) by
where and are arbitrary constants. We have a neighboring curve of the family when for , we write , . Then the function becomes
Hence, when is taken very small, it follows that
is a solution of , since it is a solution of and of .
Now we may always choose normal displacements which will take us from one of the curves to a neighboring curve . From this it appears that there is a relation between the differential equations and .
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 , also that of the differential equation is performed. We are then able to derive the general expression for , and may determine completely whether and when . We shall next derive the general solution of the equation , it being presupposed that the differential equation admits of a general solution. We derived the first variation in the form
We may form the second variation by causing in this expression alone to vary, and then alone, and by adding the results.
It follows that
Since the differential equation is supposed satisfied, we
We had (Art. 76)
When in the expression for , the substitutions
are made, we have
it follows that
When is eliminated from the last two expressions, we have
On the other hand, it is seen that
an expression which, owing to 2), 3) and 4) of the last Chapter, may be written in the following form :
and if we take into consideration 3), 4) 6) and 7) of the last Chapter, we may write the above result in the form:
In an analogous manner, we have
When these values are substituted in , we have
Hence from (a) we have
By the previous method we found the second variation to be [see formula 8) of the last Chapter]
These two expressions should agree as to a constant term. The difference of the integrals is
it is seen that
The formula (b) is
When we compare this with of the preceding Chapter, the differential equation for <maht>u</math>, viz.:
it is seen that as soon as we find a quantity for which , we have a corresponding integral of the diflEerential equation for .
The total variation of is
where , as found in the preceding article, has the value
Suppose that the equation is integrable, and let
be general expressions which satisfy it, where , are arbitrary constants of integration. The difEerential equation will be satisfied, if we suppose that and , having arbitrarily fixed values, are increased by two arbitrarily small quantities and ; that is, the functions
are also solutions of .
Now choose the variation of the curve (Art. Ill) in such a way that
and, whatever be the values of and , we determine ,,,, etc., by the relations:
For all values of and the difEerential equation satisfied; hence, the values of , , etc., just written, when substituted in above must make the right-hand side of that equation vanish identically, and consequently also . Hence, the corresponding normal displacement transforms one of the system of curves to another one of the same system.
Since and are entirely arbitrary, the coeflEcients of and must each vanish in the expansion of above. Owing to (iii) becomes
Writing this value of in the equation , we have
By equating the coefficients of and respectively to zero, we have the two equations:
where, for brevity, we have written
It is seen at once that and are the solutions of the differential equation
Hence it is seen that the general solution of the differential equation for is had from the integrals of the differential equation , through simple differentiation.
We have next to prove that the two solutions and are independent of each other. In order to make this proof as simple as possible, let be written for the arbitrary quantity .
Then the expressions , , etc., become
If and are linearly dependent upon each other, we must have
from which it follows, at once, that
where the accents denote differentiation with respect to ; or,
On the other hand, is the complete solution of the differential equation, which arises out of , when is written for ; that is, of
but here and are two arbitrary independent constants, and consequently and are independent of each other with respect to and , so that the determinant
is different from zero. Consequently and 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 , is of the form
where and are arbitrary constants.
Following the methods of Weierstrass we have just proved the assertion of Jacobi ; since, as soon as we have the complete integral of , it is easy to express the complete solution of the differential equation .
The constants and may be so determined that vanishes on a definite position , which may lie somewhere on the curve before we get to . This may be effected by writing
The solution of the equation becomes
It may turn out that vanishes for no other value of ; but it may also happen that there are other positions than at which becomes zero. If is the first zero position of which follows then is called the conjugate point to .
Since 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 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 , and nowhere vanishes within the interval .
Let the point be a zero position of the function
and let be a conjugate point to , then will not again vanish within the interval . Take in the neighborhood of the point a point , where , then the point which is conjugate to can lie only on the other side of . This may be shown as follows:
If u = \Theta(t,t') is a solution of the equation
a solution of the same equation ; that is, of
since differs from only through another choice of the arbitrary constants and .
If is chosen sujBciently small, then is different from zero and consequently also .
Eliminate from the two equations above, and we have
and the above equation becomes
which, when integrated, is
The constant in this expression cannot vanish, for, in that case,
Since, however, vanishes for , it results from the above that , which is contrary to the hypothesis, and consequently cannot vanish.
It is further assumed that does not change its sign or become zero within the interval . If vanishes without a transition from the positive to the negative or vice versa within the stretch 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, has a finite value, and the equation 7), when divided through by becomes
an expression, which, when integrated, is
Since the function does not vanish between and , it follows from the last expression that cannot vanish between the limits and . Accordingly, if there is a point conjugate to , it cannot lie before . If, therefore, we choose a point before and as close to it as we wish, then will certainly not vanish within the interval .
If is a point situated immediately before , and if we determine the point conjugate to , and choose a point before 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 , the boundaries excluded. We may then, as shown above, find a function , which satisfies the differential equation and which vanishes neither on the limits nor within the interval . The transformation of Art. 117 is therefore admissible, and the sign of depends only upon the sign of .
We may investigate a little more closely the relation of Art. 120, where
In the interval under consideration, boundaries included, we assume that does not become zero or infinite, and consequently retains the same sign. Further, the constant has always the same value and is different from zero, since and are linearly independent.
It follows at once that cannot be zero at the same time that is zero; for then would be zero contrary to our hypothesis.
Owing to the form
it is clear that has the same sign as . We may take this sign positive, since otherwise owing to the expression
we would would have positive. We may assume then that the indices have been placed upon the 's, so that is always on the increase with increasing t.
The ratio will become infinite for the zero values of (see Art. 120). Since this quotient is always increasing with increasing values of , the trace of the corresponding curve must pass through , and return again (if it does return) from . Values of , 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 , say and , which make . The abscissae are values of , and the ordinates are the corresponding values of the ratio .
To summarize : We have supposed the cases excluded in which 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 and are everywhere finite along our curve (otherwise the expression for the second variation, viz. ”
would have no meaning).
We also derived in Art. 124 the variation of in the form
and when this is compared with the differential equation
(see Art. 118),
it is seen that if an integral of the differential equation vanishes for any value of , the corresponding integral of the equation vanishes for the same value of .
In Art. 126 we had
where the displacement , takes us from a point of the curve to a point of the curve . Consequently the normal displacement can be zero only at a point where the curves and intersect.
At such a point we must have
When one of the family of curves has been selected, the two associated constants and are fixed. These are the constants that occur in and . If , further, the curve passes through a fixed point , the variable is determined, and consequently the functions and are definitely determined, so that the ratio is definitely known from the above relation. There may be a second point at which the curves and intersect. This point is the point conjugate to (see Art. 128).
The geometrical significance of these conjugate points is more fully considered in Chapter XI. Writing the second variation in the form
we see that the possibility of is when . Now is zero at both of the end-points of the curve, since at these points there is no variation, but is equal to zero at only when is conjugate to . Hence, unless the two curves and \delta G = 0 intersect again at , is not equal to zero at , and consequently
In this case, if has a positive sign throughout the interval , there is a possibility of a minimum value of the integral , and there is a possibility of a maximum value when has a negative sign throughout this interval.
Next, let be conjugate to , so that at both of the limits of integration we have . We may then take at all other points of the curve, so that consequently
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 and , and let these points be and . We may then make a displacement of the curve so that
from to ,
from to and
from to ,
where is an indeterminate constant. The quantity is subjected only to the condition that it must be zero at and , and must be a solution of the difEerential equation , and is zero at the conjugate points and .
The second variation takes the form
In the preceding article we saw (cf. also Art. 117) that
and we may therefore write in the form
where is a finite quantity.
may be written
and since, in virtue of the formula of Art. 118, the expression under this latter integral sign is zero, it follows that
Further, by hypothesis, retains the same sign within the interval , and does not become zero within or at these limits, the function is different from zero at the limits (Arts. 130 and 152), and of opposite sign at these limits, since , always retaining the same sign, leaves the value zero at one limit and approaches it at the other limit. Consequently is finite and of opposite signs at the two points and , and it remains only that be chosen finite and with the same sign, so that be different from zero. Hence by the proper choice of we may effect displacements for which 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."
↑It is sometimes possible to establish the existence or the non-existence of a maximum or a minimum by other methods ; for example, the non-existence of a minimum is seen in Case II of Art. 58. In a very instructive paper (Trans, of the Am. Math. Soc, Vol. II, p. 166) Prof. Osgood has shown that there is a minimum in the case of the g-eodesics on an ellipsoid of revolution (due to the fact that the curve must lie on the ellipsoid). Prof. Osgood says (p. 166) that Kneser's Theorem "to the effect that there is not a minimum" is in general true. It seems that each separate case must be examined for itself, and in general nothing can be said regarding a maximum or a minimum.