AQPB in which the varied curve is described. Then
General the contour consisting of the stationary curve A'Q,
express from A' to Q, the varied curve QP, front Q to P, and
vtarin for
iatlon the stationary curve A'P, from P to A', is the .boundary ofan of a cell (fig. 4). Let us denote the integral of F Integral. taken along a stationary curve by round brackets, thus
(A'Q), and the integral of F taken along any other curve by square brackets, thus [PQ]. If the varied curve is divided into a number of arcs such as QP we have the result
[AQPB]—(AB) = ~[ (A'Q) +[QP] — (A'P) 1,
and the righthand member can be expressed as a line integral taken along the varied curve AQPB.
To effect this transformation we seek an approximate expression for the term (A'Q) +[QP] _ (A'P) when Q, P are near together. Let As denote the arc QP, and 1G the angle which the tangent at P to the varied curve, in the sense from A to B, makes with the axis of x (fig. 5). Also let di be the angle which the tangent at P to the stationary curve A'P, in the sense from A' to P, makes with x the axis of x. We evaluate (A'Q)
— (A'P) approximately by means
of a result which we obtained in
connexion with the problem of
variable limits. Observing that
the angle here denoted by 4, is it equivalent to the angle formerly denoted by +w (cf. fig. t), while tan ¢ is equivalent to the quantity formerly denoted by y', we obtain the approximate equation
(A'Q) — (A'P) = —As.cos ¢ F(x,y,p)+(tan Ili— p)a } p=tan (ji
which is correct to the first order in As. Also we have
[QP] =0s . cos 4,F(x,y, tan +,)
correctly to the same order. Hence we find that, correctly to the first order in As,
(A'Q)+[QP]—(A'P)=E(x,y, tan 0, tan 4,)Os,
where
E(x, y, tan 4,, tan IG)
=cos , F(x,y, tan ¢) —F(x,y,p) — (tan tt— p) tan $.
When the parametric method is used the function E takes the form
(oaf +/zay)=,5,=w—(oaf + a )z=l,5, =m
where X, si are the direction cosines of the tangent at P to the curve AQPB, in the sense from A to B, and 1, m are the direction cosines of the tangent at P to the stationary curve A'P, in the sense from A' to P.
The function E, here introduced, has been called Weierstrass's excess function. We learn that the variation of the integral, that
Wekr is to say, the excess of the integral of F taken along the
stress's varied curve above the integral of F taken along the
excess original curve, is expressible as the line integral fEds
function. taken along the varied curve. We can therefore state a
sufficient (but not necessary) condition for the existence of an extremum in the form: When the integral is taken along astationary curve, and there is no pair of conjugate points on the arc of the curve terminated by the given end points, the integral is certainly an extremum if the excess function has the same sign at all points of a finite area containing the whole of this arc within Sufflctevt it. Further, we may specialize the excess function by
identifying A' with A, and calculating the function for a and point P on the arc AB of the stationary curve AB, and an necessary
On
arbitrary direction of the tangent at P to the varied curve. dCltlons. This process is equivalent to the introduction of a particular
type of strong variation. We may in fact take, as a varied curve, the arc AQ of a neighbouring stationary curve, the straight line QP drawn from Q to a point of the arc AB, and the arc PB of the stationary curve AB (fig. 6). The sign of the variation is then the same as that of the
function E(x, y, tan 4,, tan ¢), where (x, y) is the point P, 4, is the angle which the straight A
line QP makes with the FIG. 6.
axis of x, and di is the angle which the tangent at P to the curve APB makes with the same axis. We thus arrive at a new necessary (but not sufficient) condition for the existence of an extremum of the integral fFds, viz. the specialized excess function, so calculated, must not change sign between A and B.
The sufficient condition, and the new necessary condition, associated with the excess function, as well as the expression for the variation as fEds, are due to Weierstrass. In applications to special problems it is generally permissible to identify A' with A, and to regard QP as straight. The direction of QP must be such that the integral of F taken along it is finite and real. We shall describe such directions as admissible. In the statement of the sufficient condition, and the new necessary condition, it is of course understood that the direction specified by 4, is admissible. The excess function generally vanishes if 4,=0, but it does not change sign. It can be shown without difficulty that, when \ is very nearly equal to the sign of E is the same as that of
tan 1/.tan (a2F '2
( /
~)2 cos ay y,=tan
and thus the necessary condition as to the sign of the excess function includes Legendre's condition as to the sign of a1F/3y'2. Weierstrass's conditions have been obtained by D. Hilbert from the observation that, if p is a function of x and y, the integral
f) F(x,y,p)+(y'—p) (ay) y'=n f dx,
taken along a curve joining two fixed points, has the same value for all such curves, provided that there is a field of stationary curves, and that p is the gradient at the point (x, y) of that stationary curve of the field which passes through this point.
An instructive example of the excess function, and the conditions connected with it, is afforded by the integral
f y2y'—2dx or f y2x3y2d®.
The first integral of the principal equation is example
y2x2,r =coast., of the
and the stationary curves include the axis of x, straightlines excess parallel to the axis of y, and the family of exponentialcurves f°°ctioa. y=ae". A field of stationary curves is expressed by the equation
y=yo exp {c(x—xo)},
and, as these have no envelope other than the initial point (xo, yo), there are no conjugate points. The function fl is 6xy4, and this is positive for curves going from the initial point in the positive direction of the axis of x. The value of the excess function is
y2cos ,f (cote¢—3 coed) +2 tan coed)).
The directions 4,=o and '4, = ir are inadmissible. On putting+,=2ir we get 2y2cot3¢; and on putting >G= fir we get — 2y2cot34,. Hence the integral taken along AQ'PB is greater than that taken along APB, and the integral taken along AQPB is less than that taken along APB, when Q'Q are sufficiently near to P on the ordinate of P (fig. 7). It follows that the
integral is neither a maximum eY nor a minimum. i
It has been proved by Weierstrass that the excess function cannot be onesigned if the function f of the parametric method is a rational function of x and y. This result includes the above example, and the problem of
the solid of least resistance, `
for which, as Legendre had FIG. 7.
seen, there can be no solu
tion if strong variations are admitted. As another example of the calculation of excess functions, it may be noted that the value of the excess function in the problem of the catenoid is
2y sin2i(SG—~)•
Developments connected with the excess function.
In general it is not necessary that a field of stationary curves should consist of curves which pass through a fixed point. Any
Field family of stationary curves depending on a single para
of sta meter may constitute a field. This remark is of im
tionary portance in connexion with the adaptation of Weierstrass's
curves results to the problem of variable limits. For the purpose
and of this adaptation A. Kneser (1900) introduced the family
trans of stationary curves which are cut transversely by an
versals. assigned curve. Within the field of these curves we can
construct the transversals of the family ; that is to say, there is a finite area of the plane, through any point of which there passes one stationary curve of the field and one curve which cuts all the stationary curves of the field transversely. These curves provide a system of curvilinear coordinates, in terms of which the value of fFdx, taken along any curve within the area, can be expressed. The value of the integral is the same for all arcs of stationary curves of the field which are intercepted between any two assigned transversals.
In the above discussion of the First Problem it has been assumed that the curve which yields an extremum is an arc of a single curve, which must be a stationary curve. It is conceivable that the required curve might be made up of a finite number of arcs of different stationary curves meeting each other at finite angles. It can be shown that such a broken curve cannot yield an extremum unless both the expressions aF/ay' and F—y'(aF/ay') are continuous at the corners. In the parametric method of/ax and of/ay must be continuous at the corners. This result limits very considerably
Discon. the possibility of such discontinuous solutions, though it
t/nuous does not exclude them. An example is afforded by the
solutions. problem of the catenoid. The axis of x and any lines
parallel to the axis of y satisfy the principal equation;
and the conditions here stated show that the only discontinuous
solution of the problem is presented by the broken line ACDB
(fig. 8). A broken line
like AA'B'B is excluded.
Discontinuous solutions
have generally been sup
posed to be of special im
portance in cases where
the required curve is re
stricted by the condition
C D of not crossing the boun
axis of x dary of a certain limited
of the boundary may
have to be taken as part of the curve. Problems of this kind were investigated in detail by J. Steiner and I. Todhunter. In recent times the theory has been much extended by C. Caratheodory.
In any problem of the calculus of variations the first step is the formation of the principal equation or equations; and the second Exist step is the solution of the equation or equations, in accord
ance with the assigned terminal or boundary conditions. erica If this solution cannot be effected, the methods of the theorems. calculus fail to answer the question of the existence or nonexistence of a solution which would yield a maximum or minimum of the integral under consideration. On the other hand, if the existence of the extremum could be established independently, the existence of a solution of the principal equation, which would also satisfy the boundary conditions, would be proved. The most famous example of such an existencetheorem is Dirichlet's principle, according to which there exists a function V, which satisfies the equation
a2V a2V a2V _
art ay + a:.2 — °
at all points within a closed surface S, and assumes a given value at each point of S. The differential equation is the principal equation answering to (theeiintegral (V1
—JJJ \ax)2+(a ) 2+\a.:l2}dxdydz
taken through the volume within the surface S. The theorem of the existence of V is of importance in all those branches of mathematical physics in which use is made of a potential function, satisfying Laplace's equation; and the twodimensional form of the theorem is of fundamental importance in the theory of functions of a complex variable. It has been proposed to establish the existence of V by means of the argument that, since I cannot be negative, there must
Dirkh be, among the functions which have the prescribed
let's boundary values, some one which gives to I the smallest
principe. bysWeie stra s. He observed that precisely the same argument would apply to the integral fx2y'1dx taken along a curve from the point (—1, a) to the point (1, h). On the one hand, the principal equation answering to this integral can be solved, and it can be proved that it cannot be satisfied by any function y at all points of the interval —I 

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