THE CALCULUS OF VARIATIONS.
In order to grasp with more ease the philosophical character of the
_Method of Variations_, it will be well to begin by considering in a
summary manner the special nature of the problems, the general
resolution of which has rendered necessary the formation of this
hyper-transcendental analysis. It is still too near its origin, and its
applications have been too few, to allow us to obtain a sufficiently
clear general idea of it from a purely abstract exposition of its
fundamental theory.
PROBLEMS GIVING RISE TO IT.
The mathematical questions which have given birth to the _Calculus of
Variations_ consist generally in the investigation of the _maxima_ and
_minima_ of certain indeterminate integral formulas, which express the
analytical law of such or such a phenomenon of geometry or mechanics,
considered independently of any particular subject. Geometers for a long
time designated all the questions of this character by the common name
of _Isoperimetrical Problems_, which, however, is really suitable to
only the smallest number of them.
_Ordinary Questions of Maxima and Minima._ In the common theory of
_maxima_ and _minima_, it is proposed to discover, with reference to a
given function of one or more variables, what particular values must be
assigned to these variables, in order that the corresponding value of
the proposed function may be a _maximum_ or a _minimum_ with respect to
those values which immediately precede and follow it; that is, properly
speaking, we seek to know at what instant the function ceases to
increase and commences to decrease, or reciprocally. The differential
calculus is perfectly sufficient, as we know, for the general resolution
of this class of questions, by showing that the values of the different
variables, which suit either the maximum or minimum, must always reduce
to zero the different first derivatives of the given function, taken
separately with reference to each independent variable, and by
indicating, moreover, a suitable characteristic for distinguishing the
maximum from the minimum; consisting, in the case of a function of a
single variable, for example, in the derived function of the second
order taking a negative value for the maximum, and a positive value for
the minimum. Such are the well-known fundamental conditions belonging to
the greatest number of cases.
_A new Class of Questions._ The construction of this general theory
having necessarily destroyed the chief interest which questions of this
kind had for geometers, they almost immediately rose to the
consideration of a new order of problems, at once much more important
and of much greater difficulty--those of _isoperimeters_. It is, then,
no longer _the values of the variables_ belonging to the maximum or the
minimum of a given function that it is required to determine. It is _the
form of the function itself_ which is required to be discovered, from
the condition of the maximum or of the minimum of a certain definite
integral, merely indicated, which depends upon that function.
_Solid of least Resistance._ The oldest question of this nature is that
of _the solid of least resistance_, treated by Newton in the second book
of the Principia, in which he determines what ought to be the meridian
curve of a solid of revolution, in order that the resistance experienced
by that body in the direction of its axis may be the least possible. But
the course pursued by Newton, from the nature of his special method of
transcendental analysis, had not a character sufficiently simple,
sufficiently general, and especially sufficiently analytical, to attract
geometers to this new order of problems. To effect this, the application
of the infinitesimal method was needed; and this was done, in 1695, by
John Bernouilli, in proposing the celebrated problem of the
_Brachystochrone_.
This problem, which afterwards suggested such a long series of analogous
questions, consists in determining the curve which a heavy body must
follow in order to descend from one point to another in the shortest
possible time. Limiting the conditions to the simple fall in a vacuum,
the only case which was at first considered, it is easily found that the
required curve must be a reversed cycloid with a horizontal base, and
with its origin at the highest point. But the question may become
singularly complicated, either by taking into account the resistance of
the medium, or the change in the intensity of gravity.
_Isoperimeters._ Although this new class of problems was in the first
place furnished by mechanics, it is in geometry that the principal
investigations of this character were subsequently made. Thus it was
proposed to discover which, among all the curves of the same contour
traced between two given points, is that whose area is a maximum or
minimum, whence has come the name of _Problem of Isoperimeters_; or it
was required that the maximum or minimum should belong to the surface
produced by the revolution of the required curve about an axis, or to
the corresponding volume; in other cases, it was the vertical height of
the center of gravity of the unknown curve, or of the surface and of the
volume which it might generate, which was to become a maximum or
minimum, &c. Finally, these problems were varied and complicated almost
to infinity by the Bernouillis, by Taylor, and especially by Euler,
before Lagrange reduced their solution to an abstract and entirely
general method, the discovery of which has put a stop to the enthusiasm
of geometers for such an order of inquiries. This is not the place for
tracing the history of this subject. I have only enumerated some of the
simplest principal questions, in order to render apparent the original
general object of the method of variations.
_Analytical Nature of these Problems._ We see that all these problems,
considered in an analytical point of view, consist, by their nature, in
determining what form a certain unknown function of one or more
variables ought to have, in order that such or such an integral,
dependent upon that function, shall have, within assigned limits, a
value which is a maximum or a minimum with respect to all those which it
would take if the required function had any other form whatever.
Thus, for example, in the problem of the _brachystochrone_, it is well
known that if _y_ = _f(z)_, _x_ = π(_z_), are the rectilinear equations
of the required curve, supposing the axes of _x_ and of _y_ to be
horizontal, and the axis of _z_ to be vertical, the time of the fall of
a heavy body in that curve from the point whose ordinate is _z₁_, to
that whose ordinate is _z₂_, is expressed in general terms by the
definite integral
∫_{_z₂_}^{_z₁_}√(1 + (_f'(z))²_ + (π'(_z_))²/(2_gz_))_dz._
It is, then, necessary to find what the two unknown functions _f_ and π
must be, in order that this integral may be a minimum.
In the same way, to demand what is the curve among all plane
isoperimetrical curves, which includes the greatest area, is the same
thing as to propose to find, among all the functions _f(x)_ which can
give a certain constant value to the integral
∫_dx_√(1 + (_f'(x)_ )²),
that one which renders the integral ∫_f(x)dx_, taken between the same
limits, a maximum. It is evidently always so in other questions of this
class.
_Methods of the older Geometers._ In the solutions which geometers
before Lagrange gave of these problems, they proposed, in substance, to
reduce them to the ordinary theory of maxima and minima. But the means
employed to effect this transformation consisted in special simple
artifices peculiar to each case, and the discovery of which did not
admit of invariable and certain rules, so that every really new question
constantly reproduced analogous difficulties, without the solutions
previously obtained being really of any essential aid, otherwise than by
their discipline and training of the mind. In a word, this branch of
mathematics presented, then, the necessary imperfection which always
exists when the part common to all questions of the same class has not
yet been distinctly grasped in order to be treated in an abstract and
thenceforth general manner.
METHOD OF LAGRANGE.
Lagrange, in endeavouring to bring all the different problems of
isoperimeters to depend upon a common analysis, organized into a
distinct calculus, was led to conceive a new kind of differentiation, to
which he has applied the characteristic δ, reserving the characteristic
_d_ for the common differentials. These differentials of a new species,
which he has designated under the name of _Variations_, consist of the
infinitely small increments which the integrals receive, not by virtue
of analogous increments on the part of the corresponding variables, as
in the ordinary transcendental analysis, but by supposing that the
_form_ of the function placed under the sign of integration undergoes an
infinitely small change. This distinction is easily conceived with
reference to curves, in which we see the ordinate, or any other variable
of the curve, admit of two sorts of differentials, evidently very
different, according as we pass from one point to another infinitely
near it on the same curve, or to the corresponding point of the
infinitely near curve produced by a certain determinate modification of
the first curve.[11] It is moreover clear, that the relative
_variations_ of different magnitudes connected with each other by any
laws whatever are calculated, all but the characteristic, almost exactly
in the same manner as the differentials. Finally, from the general
notion of _variations_ are in like manner deduced the fundamental
principles of the algorithm proper to this method, consisting simply in
the evidently permissible liberty of transposing at will the
characteristics specially appropriated to variations, before or after
those which correspond to the ordinary differentials.
[Footnote 11: Leibnitz had already considered the comparison of one
curve with an other infinitely near to it, calling it
"_Differentiatio de curva in curvam_." But this comparison had no
analogy with the conception of Lagrange, the curves of Leibnitz
being embraced in the same general equation, from which they were
deduced by the simple change of an arbitrary constant.]
This abstract conception having been once formed, Lagrange was able to
reduce with ease, and in the most general manner, all the problems of
_Isoperimeters_ to the simple ordinary theory of _maxima_ and _minima_.
To obtain a clear idea of this great and happy transformation, we must
previously consider an essential distinction which arises in the
different questions of isoperimeters.
_Two Classes of Questions._ These investigations must, in fact, be
divided into two general classes, according as the maxima and minima
demanded are _absolute_ or _relative_, to employ the abridged
expressions of geometers.
_Questions of the first Class._ The _first case_ is that in which the
indeterminate definite integrals, the maximum or minimum of which is
sought, are not subjected, by the nature of the problem, to any
condition; as happens, for example, in the problem of the
_brachystochrone_, in which the choice is to be made between all
imaginable curves. The _second_ case takes place when, on the contrary,
the variable integrals can vary only according to certain conditions,
which usually consist in other definite integrals (which depend, in like
manner, upon the required functions) always retaining the same given
value; as, for example, in all the geometrical questions relating to
real _isoperimetrical_ figures, and in which, by the nature of the
problem, the integral relating to the length of the curve, or to the
area of the surface, must remain constant during the variation of that
integral which is the object of the proposed investigation.
The _Calculus of Variations_ gives immediately the general solution of
questions of the former class; for it evidently follows, from the
ordinary theory of maxima and minima, that the required relation must
reduce to zero the _variation_ of the proposed integral with reference
to each independent variable; which gives the condition common to both
the maximum and the minimum: and, as a characteristic for distinguishing
the one from the other, that the variation of the second order of the
same integral must be negative for the maximum and positive for the
minimum. Thus, for example, in the problem of the brachystochrone, we
will have, in order to determine the nature of the curve sought, the
equation of condition
δ∫_{_z₂_}^{_z₁_}√([1 + (_f'(z)_)² + (π'(_z_))²]/(2_gz_))_dz_ = 0,
which, being decomposed into two, with respect to the two unknown
functions _f_ and π, which are independent of each other, will
completely express the analytical definition of the required curve. The
only difficulty peculiar to this new analysis consists in the
elimination of the characteristic δ, for which the calculus of
variations furnishes invariable and complete rules, founded, in general,
on the method of "integration by parts," from which Lagrange has thus
derived immense advantage. The constant object of this first analytical
elaboration (which this is not the place for treating in detail) is to
arrive at real differential equations, which can always be done; and
thereby the question comes under the ordinary transcendental analysis,
which furnishes the solution, at least so far as to reduce it to pure
algebra if the integration can be effected. The general object of the
method of variations is to effect this transformation, for which
Lagrange has established rules, which are simple, invariable, and
certain of success.
_Equations of Limits._ Among the greatest special advantages of the
method of variations, compared with the previous isolated solutions of
isoperimetrical problems, is the important consideration of what
Lagrange calls _Equations of Limits_, which were entirely neglected
before him, though without them the greater part of the particular
solutions remained necessarily incomplete. When the limits of the
proposed integrals are to be fixed, their variations being zero, there
is no occasion for noticing them. But it is no longer so when these
limits, instead of being rigorously invariable, are only subjected to
certain conditions; as, for example, if the two points between which the
required curve is to be traced are not fixed, and have only to remain
upon given lines or surfaces. Then it is necessary to pay attention to
the variation of their co-ordinates, and to establish between them the
relations which correspond to the equations of these lines or of these
surfaces.
_A more general consideration._ This essential consideration is only the
final complement of a more general and more important consideration
relative to the variations of different independent variables. If these
variables are really independent of one another, as when we compare
together all the imaginable curves susceptible of being traced between
two points, it will be the same with their variations, and,
consequently, the terms relating to each of these variations will have
to be separately equal to zero in the general equation which expresses
the maximum or the minimum. But if, on the contrary, we suppose the
variables to be subjected to any fixed conditions, it will be necessary
to take notice of the resulting relation between their variations, so
that the number of the equations into which this general equation is
then decomposed is always equal to only the number of the variables
which remain truly independent. It is thus, for example, that instead of
seeking for the shortest path between any two points, in choosing it
from among all possible ones, it may be proposed to find only what is
the shortest among all those which may be taken on any given surface; a
question the general solution of which forms certainly one of the most
beautiful applications of the method of variations.
_Questions of the second Class._ Problems in which such modifying
conditions are considered approach very nearly, in their nature, to the
second general class of applications of the method of variations,
characterized above as consisting in the investigation of _relative_
maxima and minima. There is, however, this essential difference between
the two cases, that in this last the modification is expressed by an
integral which depends upon the function sought, while in the other it
is designated by a finite equation which is immediately given. It is
hence apparent that the investigation of _relative_ maxima and minima is
constantly and necessarily more complicated than that of _absolute_
maxima and minima. Luckily, a very important general theory, discovered
by the genius of the great Euler before the invention of the Calculus of
Variations, gives a uniform and very simple means of making one of
these two classes of questions dependent on the other. It consists in
this, that if we add to the integral which is to be a maximum or a
minimum, a constant and indeterminate multiple of that one which, by the
nature of the problem, is to remain constant, it will be sufficient to
seek, by the general method of Lagrange above indicated, the _absolute_
maximum or minimum of this whole expression. It can be easily conceived,
indeed, that the part of the complete variation which would proceed from
the last integral must be equal to zero (because of the constant
character of this last) as well as the portion due to the first
integral, which disappears by virtue of the maximum or minimum state.
These two conditions evidently unite to produce, in that respect,
effects exactly alike.
Such is a sketch of the general manner in which the method of variation
is applied to all the different questions which compose what is called
the _Theory of Isoperimeters_. It will undoubtedly have been remarked in
this summary exposition how much use has been made in this new analysis
of the second fundamental property of the transcendental analysis
noticed in the third chapter, namely, the generality of the
infinitesimal expressions for the representation of the same geometrical
or mechanical phenomenon, in whatever body it may be considered. Upon
this generality, indeed, are founded, by their nature, all the solutions
due to the method of variations. If a single formula could not express
the length or the area of any curve whatever; if another fixed formula
could not designate the time of the fall of a heavy body, according to
whatever line it may descend, &c., how would it have been possible to
resolve questions which unavoidably require, by their nature, the
simultaneous consideration of all the cases which can be determined in
each phenomenon by the different subjects which exhibit it.
_Other Applications of this Method._ Notwithstanding the extreme
importance of the theory of isoperimeters, and though the method of
variations had at first no other object than the logical and general
solution of this order of problems, we should still have but an
incomplete idea of this beautiful analysis if we limited its destination
to this. In fact, the abstract conception of two distinct natures of
differentiation is evidently applicable not only to the cases for which
it was created, but also to all those which present, for any reason
whatever, two different manners of making the same magnitudes vary. It
is in this way that Lagrange himself has made, in his "_Méchanique
Analytique_," an extensive and important application of his calculus of
variations, by employing it to distinguish the two sorts of changes
which are naturally presented by the questions of rational mechanics for
the different points which are considered, according as we compare the
successive positions which are occupied, in virtue of its motion, by the
same point of each body in two consecutive instants, or as we pass from
one point of the body to another in the same instant. One of these
comparisons produces ordinary differentials; the other gives rise to
_variations_, which, there as every where, are only differentials taken
under a new point of view. Such is the general acceptation in which we
should conceive the Calculus of Variations, in order suitably to
appreciate the importance of this admirable logical instrument, the
most powerful that the human mind has as yet constructed.
The method of variations being only an immense extension of the general
transcendental analysis, I have no need of proving specially that it is
susceptible of being considered under the different fundamental points
of view which the calculus of indirect functions, considered as a whole,
admits of. Lagrange invented the Calculus of Variations in accordance
with the infinitesimal conception, and, indeed, long before he undertook
the general reconstruction of the transcendental analysis. When he had
executed this important reformation, he easily showed how it could also
be applied to the Calculus of Variations, which he expounded with all
the proper development, according to his theory of derivative functions.
But the more that the use of the method of variations is difficult of
comprehension, because of the higher degree of abstraction of the ideas
considered, the more necessary is it, in its application, to economize
the exertions of the mind, by adopting the most direct and rapid
analytical conception, namely, that of Leibnitz. Accordingly, Lagrange
himself has constantly preferred it in the important use which he has
made of the Calculus of Variations in his "Analytical Mechanics." In
fact, there does not exist the least hesitation in this respect among
geometers.
ITS RELATIONS TO THE ORDINARY CALCULUS.
In order to make as clear as possible the philosophical character of the
Calculus of Variations, I think that I should, in conclusion, briefly
indicate a consideration which seems to me important, and by which I can
approach it to the ordinary transcendental analysis in a higher degree
than Lagrange seems to me to have done.[12]
[Footnote 12: I propose hereafter to develop this new
consideration, in a special work upon the _Calculus of Variations_,
intended to present this hyper-transcendental analysis in a new
point of view, which I think adapted to extend its general range.]
We noticed in the preceding chapter the formation of the _calculus of
partial differences_, created by D'Alembert, as having introduced into
the transcendental analysis a new elementary idea; the notion of two
kinds of increments, distinct and independent of one another, which a
function of two variables may receive by virtue of the change of each
variable separately. It is thus that the vertical ordinate of a surface,
or any other magnitude which is referred to it, varies in two manners
which are quite distinct, and which may follow the most different laws,
according as we increase either the one or the other of the two
horizontal co-ordinates. Now such a consideration seems to me very
nearly allied, by its nature, to that which serves as the general basis
of the method of variations. This last, indeed, has in reality done
nothing but transfer to the independent variables themselves the
peculiar conception which had been already adopted for the functions of
these variables; a modification which has remarkably enlarged its use. I
think, therefore, that so far as regards merely the fundamental
conceptions, we may consider the calculus created by D'Alembert as
having established a natural and necessary transition between the
ordinary infinitesimal calculus and the calculus of variations; such a
derivation of which seems to be adapted to make the general notion more
clear and simple.
According to the different considerations indicated in this chapter, the
method of variations presents itself as the highest degree of perfection
which the analysis of indirect functions has yet attained. In its
primitive state, this last analysis presented itself as a powerful
general means of facilitating the mathematical study of natural
phenomena, by introducing, for the expression of their laws, the
consideration of auxiliary magnitudes, chosen in such a manner that
their relations are necessarily more simple and more easy to obtain than
those of the direct magnitudes. But the formation of these differential
equations was not supposed to admit of any general and abstract rules.
Now the Analysis of Variations, considered in the most philosophical
point of view, may be regarded as essentially destined, by its nature,
to bring within the reach of the calculus the actual establishment of
the differential equations; for, in a great number of important and
difficult questions, such is the general effect of the _varied_
equations, which, still more _indirect_ than the simple differential
equations with respect to the special objects of the investigation, are
also much more easy to form, and from which we may then, by invariable
and complete analytical methods, the object of which is to eliminate the
new order of auxiliary infinitesimals which have been introduced, deduce
those ordinary differential equations which it would often have been
impossible to establish directly. The method of variations forms, then,
the most sublime part of that vast system of mathematical analysis,
which, setting out from the most simple elements of algebra, organizes,
by an uninterrupted succession of ideas, general methods more and more
powerful, for the study of natural philosophy, and which, in its whole,
presents the most incomparably imposing and unequivocal monument of the
power of the human intellect.
We must, however, also admit that the conceptions which are habitually
considered in the method of variations being, by their nature, more
indirect, more general, and especially more abstract than all others,
the employment of such a method exacts necessarily and continuously the
highest known degree of intellectual exertion, in order never to lose
sight of the precise object of the investigation, in following
reasonings which offer to the mind such uncertain resting-places, and in
which signs are of scarcely any assistance. We must undoubtedly
attribute in a great degree to this difficulty the little real use which
geometers, with the exception of Lagrange, have as yet made of such an
admirable conception.