The implicit function theorem and its substitutes in

The implicit function theorem and its substitutes
in Poincar´e’s qualitative theory of differential
equations∗
Jean Mawhin
Universit´e Catholique de Louvain
Institut de Recherche en Math´ematique et Physique
Abstract
We analyze the role of the implicit function theorem and some of its
substitutes in the work of Henri Poincar´e. Special emphasis is given upon
his PhD thesis, his first work on the periodic solutions of the three body
problem, his memoir crowned by King Oscar Prize and its development
in Les m´ethodes nouvelles de la m´ecanique c´eleste, and finally in his contributions on the figures of equilibrium of rotating fluid masses.
R´
esum´
e
Nous analysons le rˆ
ole du th´eor`eme des fonctions implicites et de certains substituts dans l’oeuvre de Henri Poincar´e. L’accent est mis en
particulier sur sa th`ese de doctorat, son premier travail sur les solutions
p´eriodiques du probl`eme des trois corps, son m´emoire couronn´e par le
Prix du Roi Oscar et son d´eveloppement dans Les m´ethodes nouvelles
de la m´ecanique c´eleste, et finalement dans ses contributions aux figures
d’´equilibre d’une masse fluide en rotation.
1
Introduction
The implicit function theorem is one of the most important and versatile tools of
mathematics, not only in analysis, but in geometry as well. Although used since
the beginning of calculus, its formalization and rigorous proof had to wait for
Cauchy [4] in the analytic case, and to Dini [10] in the smooth case. Historical
information can be found in [15, 29].
This theorem and some of its generalizations have played an important role
in the work of Henri Poincar´e, from his Thesis in 1879 till his work in celestial
mechanics. Poincar´e independently reinvented what is now called the Weierstrass preparation theorem in order to extend the Cauchy-Kovalewski theorem
∗ Studies
in the History and Philosophy of Modern Physics, 2013
1
to some singular cases. In his first work on the periodic solutions of the three
body problem, he stated and indicated the proof of a theorem which will be later
proved to be equivalent to Brouwer fixed point theorem. In his first memoir on
the figures of equilibrium of rotating fluid bodies, Poincar´e defined the concept
of bifurcation points in a series of equilibria. They are essentially the points
where the implicit function theorem does not work, and Poincar´e introduced
topological and analytic tools to prove their existence. Later, and especially in
the monographs he devoted to the mentioned problems, the implicit function
theorem and some of its extensions became the fundamental tools.
The aim of this paper is to analyze those contributions, and to show that
when Poincar´e did not take the simplest way, which is often the case for pioneers,
his detours were more than worthwhile and the sophisticated tools he invented to
solve local problems became, in the hands of other mathematicians, fundamental
for the study of the corresponding global problems.
The scientific work of Poincar´e has been recently analyzed in a nice and
detailed way in the remarkable books of Gray [13] and of Verhulst [44]. For
Poincar´e’s work on the three body problem, the reference remains BarrowGreen’s monograph [1]. Those books can be usefully consulted for a more
systematic and complete description of the memoirs and monographs considered in this paper, where a “thematic” or “transversal” viewpoint has been
emphasized more than a systematic one. We think that such a viewpoint may
be useful in understanding Poincar´e’s mathematics, because of his exceptional
talent in using a definite tool in very different areas of mathematics. Such a
viewpoint has already been developed in [24] where, instead of implicit function
techniques, Kronecker’s index had been emphasized. Other mathematical tools
could be considered as well.
2
Implicit function and preparation theorems
For the reader’s convenience, we recall in this section the statements of the main
theorems which will be often mentioned in the sequel.
The first version of the implicit function theorem was stated and proved for
analytic mappings by Cauchy [4] in 1831, and summarized in [5]. If Cm is the
cartesian product of m copies of the complex plane C, and B(r) ⊂ Cm denotes
the open ball of center 0 and radius r > 0, the mapping
F : B(r0 ) × B(R0 ) ⊂ Cn × Cp → Cp
is called analytic if it is equal on B(r0 ) × B(R0 ) to the sum of its Taylor series.
Theorem 1 If F : B(r0 ) × B(R0 ) ⊂ Cn × Cp → Cp is analytic and such that
F (0, 0) = 0, Jac y F (0, 0) 6= 0,
2
then there exists r1 ∈ (0, r0 ), R1 ∈ (0, R0 ), and f : B(r1 ) → B(R1 ) analytic
such that, in B(r1 ) × B(R1 ),
F (x, y) = 0 ⇔ y = f (x).
Recall that the Jacobian or functional determinant Jac y F (0, 0) of F with respect to y at (0, 0) is the determinant of the complex (p × p)-matrix whose
elements are the (complex) partial derivatives ∂yj Fi (0, 0) (1 ≤ i, j ≤ p).
Let Rm denote the Euclidian space of dimension m and B(r) ⊂ Rm the open
ball of center 0 and radius r > 0. The classical implicit function theorem for
mappings of class C 1 can be stated as follows.
Theorem 2 If F : B(r0 ) × B(R0 ) ⊂ Rn × Rp → Rp is of class C 1 and such
that
F (0, 0) = 0, Jacy F (0, 0) 6= 0,
then there exists r1 ∈ (0, r0 ), R1 ∈ (0, R0 ), and f : B(r1 ) → B(R1 ) of class C 1
such that, in B(r1 ) × B(R1 ),
F (x, y) = 0 ⇔ y = f (x).
Here the Jacobian or functional determinant Jacy F (0, 0) of F with respect to
y at (0, 0) is the determinant of the real (p × p)-matrix whose elements are
thepartial derivatives ∂yj Fi (0, 0) (1 ≤ i, j ≤ p). Although implicit functions
were used much earlier, the complete statement and proof of this version of the
implicit function theorem were only given by Dini [10] in his mimeographed
lectures of analysis of 1877-78, and reproduced in the monographs of GenocchiPeano [12] and Jordan [14].
Of course, in Theorems 1 and 2, the centers 0 of the involved balls, chosen for
simplicity, can be replaced by any arbitrary point of the corresponding space.
In the special situation of Theorem 1 with n = p = 1, the following result
(k)
gives information in cases where the Jacobian vanishes. Fy denotes the k th
partial derivative with respect to y.
Theorem 3 If F : B(r0 ) × B(R0 ) ⊂ Cn × C → C is analytic and such that
F (0, 0) = Fy′ (0, 0) = . . . = Fy(m−1) (0, 0) = 0, Fy(m) (0, 0) 6= 0,
(1)
then there exists r1 ∈ (0, r0 ), R1 ∈ (0, R0 ), a0 , a1 , . . . , am−1 : B(r1 ) → C analytic, vanishing at 0, and G : B(r1 ) × B(R1 ) → B(R1 ) analytic such that
G(x, y) 6= 0 on B(r1 ) × B(R1 ) and
F (x, y) = [a0 (x) + . . . + am−1 (x)y m−1 + y m ]G(x, y) on B(r1 ) × B(R1 ).
In other words, the zeros of F (x, ·) in a neighborhood of (0, 0) are the solutions
of the algebraic equation
a0 (x) + . . . + am−1 (x)y m−1 + y m = 0.
3
This result is usually called Weierstrass preparation theorem. For m = 1, it
implies of course Theorem 1 with n = p = 1. As observed by Lindel¨
of in [16],
Cauchy [4] stated and proved it already in 1831, and published it in 1841 [5].
Weierstrass [45] stated and proved it in his Berlin’s lectures around 1860, and
published it in 1886. Poincar´e, as we shall see, stated, proved and published it
in 1879.
3
1879 : Sur les propri´
et´
es des fonctions d´
efinies par les ´
equations aux diff´
erences partielles
Poincar´e’s thesis [34], entitled Sur les propri´et´es des fonctions d´efinies par les
´equations aux diff´erences partielles, and defended in 1879, starts with some
‘preliminary lemmas”. The first one, called by Poincar´e “Th´eor`eme de BriotBouquet” is nothing but Theorem 1. The unusual name given by Poincar´e comes
from the fact that the reference he gave for this theorem is Briot-Bouquet’s
famous treatise on elliptic functions [3]. This may have pleased Bouquet, a
member of the jury. Poincar´e added that
this theorem can be seen as a consequence of the theorem of existence of
the integral of a differential equation.1
Then Poincar´e introduced the concept of an algebro¨ıd function y from C to
C, namely a function y which, in a neighborhood of 0 ∈ C, is solution of an
equation of the form
y m + Am−1 (x)y m−1 + . . . + A1 (x)y + A0 (x) = 0,
where m ≥ 1 is an integer and the Aj vanish at 0 and are analytic near 0.
P∞
If now F (x, y) := k=0 Ak (x)y k , where the analytic functions Aj of x :=
(x1 , . . . , xn ) ∈ Cn are such that
A0 (0) = . . . = Am−1 (0) = 0, Am (0) 6= 0,
(which means that F is analytic in the neighborhood of (0, 0) and condition (1)
holds), Poincar´e stated and proved the following two results as Lemma II and
Lemma III.
Lemma 1 There exist m functions y(x) such that, near 0,
F (x, y(x)) = 0 and lim y(x) = 0.
x→0
Lemma 2 The m functions y(x) are algebro¨ıd of degree m.
1 ce th´
eor`
eme peut ˆ
etre regard´
e comme une cons´
equence du th´
eor`
eme relatif a
` l’existence
de l’int´
egrale d’une ´
equation diff´
erentielle.
4
So, a third independent author must be added to Cauchy and Weierstrass for
essentially proving the preparation theorem.
In the thesis, those results are applied to the obtention of some extensions
of Briot-Bouquet theorems for singular ordinary differential equations to the
Cauchy-Kowalevski’s problem for analytic partial differential equations.
4
1883-84 : Sur certaines solutions particuli`
eres du probl`
eme des trois corps
Poincar´e’s first paper on the three body problem was published in 1884 [38] and
anounced in 1883 in a note to the Comptes rendus [37].
Poincar´e considered the motion of three material points with respective
masses m, m′ , M , in the situation where m/M and m′ /M are very small. He was
interested by showing the existence of periodic solutions, i.e. solutions which
return to the same position, with the same speed, after some period of time, in a
suitable reference rotating system with respect to an inertial one. Furthermore,
those periodic solutions were requested to satisfy in addition some symmetry
properties that we do not describe here.
After a number of transformations, the problem was reduced by Poincar´e to
finding the common zeros of three analytic functions X, Y, Z of the masses m, m′
(assumed to be small by taking M = 1) and of the six initial elements in a suitable system of astronomical coordinates. Poincar´e wrote X = X(m, m′ ; x, y, z)
in the form
X = X0 + X1 + X2 + . . . + Xn + . . . ,
(2)
where Xn is of the nth order in (m, m′ ), and similarly for Y, Z.
Taking three of the initial elements constant, Poincar´e assumed that the
three other ones (x, y, z) could be chosen, after a translation, such that,
X0 (0, 0; 0; 0, 0) = Y0 (0, 0; 0, 0, 0) = Z0 (0, 0; 0, 0, 0) = 0,
Jac(x,y,z) (X0 , Y0 , Z0 )(0, 0; 0, 0, 0) 6= 0.
He then claimed that, if it is the case, nearby initial elements (x, y, z) can be
chosen so that, for m and m′ small,
X(m, m′ ; x, y, z) = Y (m, m′ , x, y, z) = Z(m, m′ ; x, y, z) = 0,
so that (x, y, z) are the initial conditions of periodic solutions for the small
masses m, m′ .
The reader can immediately observe that this claim follows immediately from
Theorem 2, which was well known to Poincar´e. Surprisingly he took another
5
complicated way to prove it. He started by observing that there exist small
positive numbers a1 , a2 , a3 such that the points
(X0 (0, 0; x, y, z), Y0 (0, 0; x, y, z), Z0 (0, 0; x, y, z))
cover the parallelotope made of the (X0 , Y0 , Z0 ) such that
X02 < a21 , Y02 < a22 , Z02 < a23 ,
if one takes the initial elements (x, y, z) in a suitable neighborhood of (0, 0, 0). In
modern terms, (X0 (0, 0; ·), Y0 (0, 0; ·), Z0 (0, 0; ·)) is an open mapping at (0, 0, 0).
This is a consequence of the inverse mapping theorem, itself equivalent to the
implicit function theorem !
This allowed Poincar´e to consider X, Y, Z as functions of the masses m, m′
and of (X0 , Y0 , Z0 ). Now, for m, m′ small with respect to a1 , a2 , a3 , it follows
immediately from (2) that
X > 0 for X0 = a1 , X < 0 for X0 = −a1 ,
and similarly for Y, Z. From this, Poincar´e concluded that, for sufficiently small
masses m, m′ , some (X0 , Y0 , Z0 ) (and hence initial elements (x, y, z)) exist, such
that X = Y = Z = 0, by using the following existence theorem.
Theorem 4 Let X1 , X2 , . . . , Xn be continuous functions of x1 , x2 , . . . , xn such
that
Xi > 0 for xi = ai and Xi < 0 for xi = −ai (i = 1, . . . , n).
Then there exists x1 , x2 , . . . , xn such that |x1 | < a1 , |x2 | < a2 , . . . |xn | < an and
X1 (x1 , . . . , xn ) = 0, X2 (x1 , . . . , xn ) = 0, . . . , Xn (x1 , . . . , xn ) = 0.
Geometrically, if, for each i = 1, . . . , n, the ith component of X = (X1 , . . . , Xn )
takes opposite signs on the opposite ith faces of the parallelotope, then X has
a zero in the parallelotope. This statement is clearly an n-dimensional generalization of Bolzano’s intermediate value theorem for real continuous functions of
one real variable.
In [37], Poincar´e gave the following heuristic proof when n = 2. For each
fixed x1 ∈ [−a1 , a1 ], the function X2 seen as a function of x2 takes opposite
signs at −a2 and a2 and hence vanishes between, giving a “curve” of zeros of
f2 joining the opposite sides x1 = −a1 and x1 = a1 of the rectangle. Similarly
one obtains a “curve” of zeros of X1 joining the opposite sides x2 = −a2 and
x2 = a2 of the rectangle. The two “curves” must meet in some point of the
rectangle, where X1 and X2 vanish simultaneously. To change this argument
into a real proof requires some algebraic topology !
Poincar´e’s rigorous proof of Theorem 4 is more than sketchy, identical in [34]
and [37], and consists in the following three lines :
6
Mr. Kronecker has presented to the Berlin Academy, in 1869, a Memoir
on functions of several variables; one can find there an important theorem
from which it is easy to deduce the following result.2
And then comes Theorem 4.
Kronecker’s memoir [17], 46 pages in two parts of hard analysis, contains
many theorems. The one mentioned by Poincar´e implies the existence of a zero
in a n-dimensional bounded domain with smooth boundary, for a system of n
smooth functions of n variables defined there and having no common zero on
the boundary, when the integral over the boundary of some expression involving
the Xi and their partial derivatives, is different from zero. This integral is now
referred as Kronecker integral or Kronecker index.
Notice that Poincar´e’s Theorem 4 deals formally with continuous functions
(which need not to be smooth) on a parallelotope (whose boundary is not
smooth). Of course, “continuous” in Poincar´e’s language could mean “smooth”,
and Kronecker’s integral makes sense on the boundary of a parallelotope. Poincar´e’s “easy deduction” must have been much less easy for his contemporaries
than for to-day’s mathematicians familiar with topological degree arguments
and homotopy invariance !
For the experts, this Kronecker integral or Kronecker index, on the smooth
boundary ∂D of a bounded domain D ⊂ Rn , of the smooth mapping X =
(X1 , . . . , Xn ) such that 0 6∈ X(∂D) is defined by
Z
1
X ∗ σ,
iK [X, ∂D] :=
µ(S n−1 ) ∂D
where the exterior differential form σ is defined by
σ :=
n
X
j=1
(−1)j−1
yj
cj ∧ dyj+1 ∧ . . . ∧ dyn ,
dy1 ∧ . . . ∧ dyj−1 ∧ dy
|y|n
cj means that
µ(S n−1 ) is the measure of the (n − 1)-dimensional unit sphere, dy
dyj is missing in the exterior product, and * denotes the pull-back operation.
If X has only simple zeros x1 , . . . , xm in D, Kronecker has shown that
iK [X, ∂D] =
m
X
sgn Jac X(xj ),
j=1
so that Kronecker index may be seen as an algebraic count of the number of
zeros of X in D. The advantage of the algebraic count with respect to the
brute count is to remain constant for sufficiently small perturbations of X. The
2 M. Kronecker a pr´
esent´
e a
` l’Acad´
emie de Berlin, en 1869, un M´
emoire sur les fonctions
de plusieurs variables; on peut y trouver un important th´
eor`
eme d’o`
u il est ais´
e de d´
eduire le
r´
esultat suivant.
7
extension of Kronecker’s index to continuous X and arbitrary bounded open
D is the Brouwer degree dB [X, D] (see e.g. [27] for a definition and historical
information).
Theorem 4, that Poincar´e will not use any more in any further work, is a
global result, by contrast to the local character of the implicit function theorem.
Soon forgotten, it has been rediscovered by Silvio Cinquini [8] in 1940, who
“proved” it by using an argument similar to Poincar´e’s heuristic one. One
year later, Carlo Miranda [30] showed in an elementary way its equivalence to
Brouwer fixed point theorem, which asserts that any continuous transformation
of a parallelotope into itself leaves at least one point fixed. Under the name of
Miranda theorem, Theorem 4 has been part of a ten years war between Cinquini
and Giuseppe Scorza-Dragoni about its nature (topology or analysis ?) and the
validity of its various “proofs”. Poincar´e’s priority was rediscovered in 1974
only, and the result renamed Poincar´e-Miranda theorem. One can consult [25]
for a detailed story and references.
Nowadays, the Poincar´e-Miranda theorem, Brouwer fixed point theorem and
degree arguments are widely used to obtain global results for the existence of
periodic solutions and boundary value problems associated to ordinary differential equations. The interested reader can find an analytic proof of Theorem
4 independent of the concept of index in [28].
Incidently, Poincar´e-Miranda’s theorem allows to prove an extension of Theorem 2 to the case where F is continuous on B(r) × B(R) and Fy′ only exists
at (0, 0), but the uniqueness of f is lost. The proof by induction of this result
given in respected textbooks on analysis (for example [9, 46]) are not correct.
More generally, Brouwer degree theory provides an implicit function theorem
(again without uniqueness) for F continuous only, when F (0, y) 6= 0 on some
sufficiently small sphere ∂B(R) and the Brouwer degree dB [F (0, ·), B(R)] is different from 0. The complex version of this generalization covers the assumptions
of the preparation theorem 3, but does not provide the factorization.
As shown in [24], Kronecker index has been widely used by Poincar´e between
1881 and 1886. The first occurrence is contained in his first memoir of 1881 on
the curves defined by a differential equation [35]. It is restricted to n = 2, and
uses an equivalent definition given by Cauchy [6] in 1837. On February 14, 1883,
a letter of Kronecker calls Poincar´e’s attention to his memoir [17] :
I would like to call your attention to a memoir that I have published
in 1869 and that I take the liberty to send you [...]. I have developed
there the generalization of this important theorem of Cauchy [...] which
seems to me to contain the true foundations of function theory. It is
very remarkable that there exists a theorem completely analogous for an
arbitrary number of variables.3
3 Je d´
esirerais appeler votre attention a
` un m´
emoire que j’ai publi´
e en 1869 et que je prend
la libert´
e de vous envoyer. [...] J’y ai d´
evelopp´
e la g´
en´
eralisation de cet important the´
eor`
eme
de Cauchy, qui me semble contenir le vrai fondement de la th´
eorie des fonctions. Il est tr`
es
8
It took a very short time to Poincar´e to assimilate Kronecker’s difficult memoir.
In a paper submitted on July 20 1883, Poincar´e applied Kronecker’s index in
dimension n to a study of the zeros of the functions Θ of several variables [36].
Three months later, the note [37] containing Theorem 4 and its application to
the three body problem was presented to the Academy ! Another application
to the equilibrium of a rotating fluid body [39], discussed later, came in 1885.
A further application in 1886 to the qualitative theory of differential equations
of higher order [40], was the first one where Poincar´e cared to give the explicit
definition of this Kronecker’s integral he has used so widely !
5
1890 : Sur le probl`
eme des trois corps et les
´
equations de la dynamique
Implicit function arguments are often used in Poincar´e’s famous memoir of 1890
[41], crowned by King Oscar Prize (see [1] for details).
Theorem IV of Part 1, Chapter 1 is the analytic implicit function theorem,
this time correctly attributed to Cauchy. When dealing in Theorem V with the
preparation theorem, Poincar´e wrote :
It would remain to examine what happens when the functional determinant of the f with respect to the y is zero. This question has been the
object of many researches, where one should first quote the work of M.
Puiseux on the roots of algebraic equations. I had myself the opportunity
to consider analogous researches in the first part of my thesis.4
Surprisingly, Poincar´e omitted here to mention the important contributions of
Weierstrass, Chairman of the jury of King Oscar prize !
Poincar´e then showed that the case of a null Jacobian in Theorem 1 could
be reduced to Theorem 3 applied to a single equation of the form Φ(x, yn ) = 0,
by successive elimination of y1 , . . . , yn−1 .
In Chapter 2, Theorem 1 is used to prove that, given an autonomous differential system
x˙ = X(x, y, z, µ), y˙ = Y (x, y, z, µ), z˙ = Z(x, y, z, µ)
(3)
depending upon a small parameter µ, a contactless surface S, and a point
P0 = (x0 , y0 , z0 ) ∈ S close to A = (a0 , b0 , c0 ) ∈ S, the next intersection
P1 = (x1 , y1 , z1 ) with S of the solution of (3) issued from P0 (the first consequent) is a analytic mapping of (x0 − a0 , y0 − b0 , z0 − c0 ) and µ.
remarquable, qu’il existe un th´
eor`
eme tout-`
a-fait analogue pour un nombre quelconque de
variables.
4 Il nous resterait a
` examiner ce qui se passe quand le d´
eterminant fonctionnel des f par
rapport aux y est nul. Cette question a fait l’objet de recherches nombreuses sur lesquelles
je ne puis insister ici, mais au premier rang desquelles il convient de citer les travaux de M.
Puiseux sur les racines des ´
equations alg´
ebriques. J’ai eu moi-mˆ
eme l’occasion de m’occuper
de recherches analogues dans la premi`
ere partie de ma Th`
ese inaugurale.
9
Chapter 3 is devoted to the study of periodic solutions of differential systems
of the form
x˙ = X(t, x, µ)
(4)
where x = (x1 , . . . , xn ), X is 2π-periodic in t (or independent of t), and system
(4) with µ = 0 has a 2π-periodic solution ϕ(t). The question raised by Poincar´e
is the following one :
Under which conditions will we be allowed to conclude that the equations
still possess periodic solutions for small values of µ ?5
For |µ| small, the solution of (4) equal to φ(0) + β at t = 0 is denoted by
x(t, µ, β). By uniqueness, x(t, 0, 0) = ϕ(t). Poincar´e observed that x(t, µ, β) is
2π-periodic if and only if β is such that x(2π, µ, β) = ϕ(0) + β, i.e. if and only
if
ψ(µ, β) := x(2π, µ, β) − ϕ(0) − β = 0.
Notice that ψ(0, 0) = 0, and hence Theorem 2 implies that if Jacβ ψ(0, 0) 6= 0,
then ψ(µ, β) = 0 has a solution β(µ) for |µ| sufficiently small, which proves the
following result.
Theorem 5 If Jacβ ψ(0, 0) 6= 0, system (4) has a 2π-periodic solution for all
sufficiently small |µ|.
In proving this result, Poincar´e replaced here the use of his n-dimensional
intermediate value theorem by the classical implicit function theorem. This
indirect disparition of Kronecker’s name may have pleased the Chairman of the
jury of King Oscar Prize.
Poincar´e’s method for finding periodic solutions of (4) that we have just
described can bring situations where the Jacobian of ψ is zero. In this case
the elimination of β1 , . . . , βn−1 as indicated above usually leads to an equation
Φ(µ, βn ) = 0, with β = 0 is a root of multiplicity m of the equation Φ(0, β) = 0.
Then (remembering that we are in the real case), Poincar´e observed that a 2πperiodic solution still exists for m odd and |µ| small, as a consequence of the
preparation theorem and the fact that a real algebraic equation of odd order
always has a real solution.
Situations where the Jacobian will always vanish occur when system (4)
with µ = 0 admits a family ϕ(t, h) of 2π-periodic solutions, or when (4) has
a first integral (for example Hamiltonian systems). This is the case when (4)
does not depend explicitely upon t. If the scale of time is chosen so that ϕ(t)
is 2π-periodic,and if x(t, µ, β) is (2π + τ )-periodic, ψ1 , . . . , ψn will depend on
the n + 1 variables β1 , . . . , βn , τ but x(t + h, µ, β) will also be (2π + τ )-periodic
for all h, which will allow in general to chose arbitrarily one of the βj and have
a non-zero Jacobian with respect to the other βj and τ .
5 Dans quelles conditions aura-t-on encore le droit d’en conclure que les ´
equations comportent encore des solutions p´
eriodiques pour les petites valeurs de µ ?
10
6
1892-1899 : Les m´
ethodes nouvelles de la m´
ecanique c´
eleste
This fundamental book [42], which develops the crowned memoir [41] into 3
volumes totalizing more than 1250 pages, contains many applications of implicit
function techniques.
Chapter 2 of Volume 1 essentially repeats the standard results on implicit
functions, the preparation and elimination theorems given in [41]. More surprisingly, Kronecker’s index reappears in the last section of this chapter, to show
that if a real smooth function F (x) := F (x1 , . . . , xn ) has a strict maximum at
0, so that
∇F (0) = 0.
(5)
then 0 is a solution of odd order of (5). To prove this result, Poincar´e considered the small closed level surface S of equation F (x) = F (0) − λ2 and a
family of functions Φ(x, µ) such that Φ(x, 0) = F (x), and the zeros ξ1 , . . . , ξm
of ∇x Φ(x, µ) are simple. Therefore, by the properties of Kronecker index,
iK [∇F, S] = 1 = iK [∇x Φ(·, µ), S]
for |µ| sufficiently small, and
1 = #{ξj : Jacx Φ > 0} − #{ξj : Jacx Φ < 0}
m = #{ξj : Jacx Φ > 0} + #{ξj : Jacx Φ < 0},
where #E denotes the number of elements of the set E. Consequently,
1 + m = 2#{ξj : Jac x Φ > 0},
and m is odd.
Chapter 3 of Volume 3 is devoted to periodic solutions of systems depending
upon a small parameter. The general theory is close to that given in [41]. Its
application to the three body problem develops the results of [37] using implicit
function techniques instead of Theorem 4.
Some special attention is paid to Hamiltonian systems depending upon a
parameter, when the unperturbed system has a zero Hessian. The study of non
trivial periodic solutions near an equilibrium 0 of systems of the form
x˙ = X(x, µ)
such that X(0, µ) = 0 for all µ again leads to an implicit function problem with
zero Jacobian, and is briefly considered. Those results of Poincar´e will inspire
further work of Gilbert A. Bliss [2], William D. McMillan [18]-[23], Forest R.
Moulton [31]-[33], Ioel G. Malkin, Earl A. Coddington and Norman Levinson,
Warren S. Loud,... and many others (see references in [7, 44]).
11
Chapter 4 is devoted to the theory of characteristic exponents. Considering
again system (4) which, for µ = 0, has a T-periodic solution ϕ(t), the variational
equation (already considered by Jacobi and Darboux in other contexts)
y˙ = Xx′ (t, ϕ(t), 0)y
(6)
is introduced. As shown by Floquet [11], the fundamental matrix solution Y (t)
of (6) has the form Y (t) = Z(t)etS , where Z(t) is T-periodic and S is a constant
matrix, whose eigenvalues are the characteristic exponents of (6). A zero characteristic exponent corresponds to a T-periodic solution of (6). Poincar´e then
proved the following existence result.
Theorem 6 If all characteristic exponents of (6) are different from zero, system
(4) has a T-periodic solution for small |µ|.
In the case of an autonomous system
x˙ = X(x, µ)
(7)
having, for µ = 0 a T-periodic solution ϕ(t), the corresponding variational
equation
y˙ = Xx′ (ϕ(t), 0)y
(8)
always has the T-periodic solution ϕ(t),
˙
and hence a characteristic exponent
equal to 0. Poincar´e proved for (7) the following variant of Theorem 6.
Theorem 7 If n − 1 characteristic exponents of (8) are different from zero,
system (7) has a periodic solution of period close to T for small |µ|.
The proofs of Theorems 6 and 7 are again based upon implicit function techniques.
Notice that Theorem 6 can be seen as an anticipation of an implicit function
theorem in the frame of infinite dimensional spaces of T-periodic functions.
Indeed, if we define the mapping F from CT1 × R into CT , where CT1 is the space
of T-periodic functions of class C 1 , and CT the space of continuous T-periodic
functions, by the formula
F (x(·), µ) := x(·)
˙ − X(·, x(·), µ),
then the Fr´echet derivative Fx′ of F at (ϕ, 0) is the linear operator from CT1 into
CT given by
Fx′ (ϕ(·), 0)y(·) = y(·)
˙ − Xx′ (·, ϕ(·), 0)y(·),
i.e. the linear operator between CT1 and CT associated to the variational equation. The conditions on the characteristic exponents in Theorem 6 correspond to
the invertibility of the Fr´echet derivative, corresponding the nonvanishing of the
Jacobian in the finite-dimensional setting. Of course, none of those functional
analytic concepts were defined when Poincar´e published [42].
12
Implicit function theorems are still used in Chapter 5 devoted to the problem
of non-existence of other uniform integrals that the classical ones in the three
body problem, in Chapter 28, 30 and 31 to study the periodic solutions of second
kind (genre) (i.e. of smallest period kT for some integer k > 1), their formation
and their properties. The reader is invited to consult [44] for a detailed analysis
of the content of [42] and its modern consequences.
7
1895 : Sur l’´
equilibre d’une masse fluide anim´
ee d’un mouvement de rotation
Starting with the ellipsoid of revolution, new figures of equilibrium of rotating
fluid bodies submitted to gravitation have been obtained, after Newton, by
MacLaurin, Jacobi, Riemann and others.
In 1885, Poincar´e [39] had the idea of considering the evolution of the various
known equilibria and possible new (pear-shaped) ones, by introducing the concept of series of equilibria with the angular velocity as parameter. He wanted
to show that new shapes of equilibrium could bifurcate from another one for
some values of the parameter, and exchange their stability. Such results were
of course important in the discussion of the evolution of planets and celestial
bodies.
To motivate his results, Poincar´e started with the simpler situation of the
evolution of the equilibria of a smooth potential F (x, y1 , y2 , . . . , yn ) depending
on a parameter x. Those equilibria satisfy of course the equations
Fy′ 1 (x, y) = . . . = Fy′ n (x, y) = 0,
and Poincar´e assumed that 0 was an equilibrium when x = 0, namely that
Fy′ 1 (0, 0) = . . . = Fy′ n (0, 0) = 0.
If the Hessian det(Fy′′j ,yk )(0, 0) 6= 0, the implicit function theorem implies
the existence of a locally unique solution, i.e. of a unique branch of equilibria
passing through (0, 0) and close to this point. Hence a necessary condition for
bifurcation, i.e. for another branch to arise at (0, 0), is that det(Fy′′j ,yk (0, 0)) = 0.
So, in this setting, bifurcation is the study of what happens when the conditions
of the implicit function theorem are not satisfied.
In the special case where n = 1 and Fy′ (0, 0) = 0, Poincar´e showed that
a sufficient condition for the existence of another branch y = f (x) such that
′′
f (0) = 0 was that ∆(x) := Fy,y
(x, f (x)) changes sign at 0.
The situation is more complicated when n = 2. The equilibrium condition
at 0 gives the system
Fy′ 1 (0, 0, 0) = Fy′ 2 (0, 0, 0) = 0.
13
Let
Fy′ (x, y1 , y2 ) := (Fy′ 1 (x, y1 , y2 ), Fy′ 2 (x, y1 , y2 )).
For x fixed, let iK [Fy′ (x, ·, ·), C] denote the Kronecker index of Fy′ (x, ·) along a
small circle C centered at (0, 0). Using once more the properties of Kronecker
index, Poincar´e has proved the following sufficient condition for bifurcation.
Theorem 8 If iK [Fy′ (−ε; ·, ·), C] 6= iK [Fy′ (+ε; ·, ·), C] for all sufficiently small
ε > 0, then (0, 0, 0) is a bifurcation point of Fy′ .
A closer look to Poincar´e’s proof shows that it remains valid for arbitrary n
and when the gradient mapping Fy′ is replaced by a general smooth one. This
result of Poincar´e clearly anticipates the modern topological approach for the
existence of bifurcation points. One can consult [26] for details and references
to contemporary literature.
As a special case of Theorem 8, Poincar´e showed that if det(Fy′′j ,yk (x, 0, 0))
changes sign at x = 0, then (0, 0, 0) is a bifurcation point of Fy′ , which generalized
the elementary result obtained for n = 1.
8
1902 : Figures d’´
equilibre d’une masse fluide
Poincar´e returned to the problem of the figures of equilibrium of rotating fluid
bodies in his lectures at the Sorbonne of 1900, written down by L. Dreyfus and
published in 1902 [43].
Again, he first considered a potential function depending upon a parameter
µ, F (x1 , . . . , xn , µ) := F (x, µ) and having, for µ = 0, an equilibrium at 0, so
that
Fx′ 1 (0, 0) = . . . = Fx′ n (0, 0) = 0.
Using Theorem 2, he observed that if detFx′′ (0, 0) 6= 0 then, the equilibrium
equations
Fx′ 1 (x, λ) = . . . = Fx′ n (x, λ) = 0
have, near (0, 0) a unique solution x = ψ(µ) for some function ψ such that
ψ(0) = 0. Hence, the condition
′′
det(Fx,x
)(0, 0) = 0
is a necessary condition for the existence of a second branch near (0, 0), i.e. for
bifurcation at (0, 0).
Using this time implicit function techniques only, Poincar´e showed that if
′′
at least one of the minors of det(Fx,x
)(0, 0) is different from zero, successive
reductions allow to write xj = ϕj (x1 , µ) for some function ϕj (2 ≤ j ≤ n).
Letting
ψ(x1 , µ) := F (x1 , ϕ2 (x1 , µ), . . . , ϕn (x1 , µ), µ),
the equilibrium condition becomes ψx′ 1 (0, 0) = 0. For this equation, a necessary
condition for bifurcation is that ψx′′1 ,x1 (0, 0) = 0, and Poincar´e discussed the
14
shape of the bifurcation in terms of ψx′′1 ,µ , and the zones of stability of the
equilibrium in the plane (x1 , µ).
9
Conclusions
A general conclusion of this analysis of the role and use of implicit function
techniques in Poincar´e’s work on the qualitative theory of differential equations,
the three body problem and the figures of equilibrium of rotating fluid bodies is
that the paths of a genius are unpredictable. Clearly, Poincar´e knew very well
the implicit function theorem before writing his thesis, and he even rediscovered
the preparation theorem independently of Cauchy and Weierstrass.
In his first papers on the periodic solutions of the three body problem and
on the figures of equilibrium of rotating fluid bodies, Poincar´e used efficiently
Kronecker index (at this time a sophisticated and little known tool) in situations
where the implicit function theorem was sufficient, as confirmed with his further
contributions in papers or monographs, where Kronecker index was abandoned
for more classical techniques.
But, this superfluous detour gave him the opportunity to create the topological approach to nonlinear differential equations and bifurcation, from which
not only local results, but also and mainly global results, outside of the scope
of implicit function theorems, can be obtained.
The paths of the Lord are claimed to be unpredictable. With Poincar´e, it is
not only often difficult to understand “how he did it this way”, but still more
difficult to know “why he did it that way”.
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18

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