The implicit function theorem and its substitutes in Poincaré׳s qualitative theory of differential equations

The implicit function theorem and its substitutes in Poincaré'squalitative theory of differential equations

Jean MawhinInstitut de Recherche en Mathématique et Physique, Université Catholique de Louvain, chemin du Cyclotron 2, B-1348 Louvain-la-Neuve, Belgium

a r t i c l e i n f o

Article history:Received 28 December 2013Accepted 13 January 2014

Keywords:PoincaréImplicit function theoremDifferential equation

a b s t r a c t

We analyze the role of the implicit function theorem and some of its substitutes in the work of HenriPoincaré. Special emphasis is given upon his PhD thesis, his first work on the periodic solutions of thethree body problem, his memoir crowned by King Oscar II Prize and its development in Les méthodesnouvelles de la mécanique céleste, and finally his contributions on the figures of equilibrium of rotatingfluid masses.Résumé: Nous analysons le rôle du théorème des fonctions implicites et de certains substituts dansl'oeuvre de Henri Poincaré. L'accent est mis en particulier sur sa thèse de doctorat, son premier travail surles solutions périodiques du problème des trois corps, son mémoire couronné par le Prix du Roi Oscar IIet son développement dans Les méthodes nouvelles de la mécanique céleste, et finalement ses contribu-tions aux figures d’équilibre d'une masse fluide en rotation.

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1. Introduction

The implicit function theorem is one of the most importantand versatile tools of mathematics, not only in analysis, but ingeometry as well. Although used since the beginning of calculus,its formalization and rigorous proof had to wait for Cauchy (1831)in the analytic case, and to Dini (1878) in the smooth case.Historical information can be found in Krantz-Parks (2002) andMingari Scarpello-Ritelli (2002).

This theorem and some of its generalizations have played animportant role in the work of Henri Poincaré, from his Thesis in1879 till his work in celestial mechanics. Poincaré independentlyreinvented what is now called the Weierstrass preparation theo-rem in order to extend the Cauchy–Kovalewski theorem to somesingular cases. In his first work on the periodic solutions of thethree body problem, he substituted to the implicit functiontheorem a topological result which will be later proved to beequivalent to Brouwer fixed point theorem. In his first memoiron the figures of equilibrium of rotating fluid bodies, Poincarédefined the concept of bifurcation points in a series of equilibria.They are essentially the points where the implicit functiontheorem does not work, and Poincaré introduced topological andanalytic tools to prove their existence. Later, and especially in the

monographs he devoted to the mentioned problems, the implicitfunction theorem and some of its consequences became thefundamental tools.

The aim of this paper is to analyze those contributions, and toshow that when Poincaré did not take the simplest way, which isoften the case for pioneers, his detours were more than worth-while and the sophisticated tools he invented to solve localproblems became, in the hands of other mathematicians, funda-mental for the study of the corresponding global problems.

The scientific work of Poincaré has been recently analyzed in anice and detailed way in the remarkable books of Gray (2012)and of Verhulst (2012). For Poincaré's work on the three bodyproblem, the reference remains Barrow-Green's (1997) mono-graph. Those books can be usefully consulted for a more systema-tic and complete description of the memoirs and monographsconsidered here.

In this paper, a “thematic” or “transversal” viewpoint isemphasized more than a systematic one. We believe that such aviewpoint may be useful in understanding and analyzing Poin-caré's mathematics, because of his exceptional talent in using adefinite tool in very different areas of mathematics. Such a view-point has already been developed in Mawhin (2000), where,instead of implicit function techniques, Kronecker's index hadbeen emphasized. Other mathematical tools could be consideredas well, like non-Euclidean geometry, group theory, calculus ofvariations or anticipations of exterior calculus for example.

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Please cite this article as: Mawhin, J. The implicit function theorem and its substitutes in Poincaré's qualitative theory ofdifferential equations. Studies in History and Philosophy of Modern Physics (2014), http://dx.doi.org/10.1016/j.shpsb.2014.01.006i

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2. Implicit function and preparation theorems

For the reader's convenience, we recall in this section thestatements of the main theorems which will be often mentionedin the sequel.

The first version of the implicit function theorem was statedand proved for analytic mappings by Cauchy (1831), and summar-ized in Cauchy (1841). If Cm is the cartesian product of m copies ofthe complex plane C, and BðrÞ �Cm denotes the open ball of center0 and radius r40, the mapping

F : Bðr0Þ � BðR0Þ �Cn � Cp-Cp

is called analytic if it is equal on Bðr0Þ � BðR0Þ to the sum of itsTaylor series.

Theorem 1. If F : Bðr0Þ � BðR0Þ �Cn � Cp-Cp is analytic and suchthat

Fð0;0Þ ¼ 0; JacyFð0;0Þa0;

then there exist r1A ð0; r0Þ, R1Að0;R0Þ, and f : Bðr1Þ-BðR1Þ analyticsuch that, in Bðr1Þ � BðR1ÞFðx; yÞ ¼ 03y¼ f ðxÞ:

Recall that the Jacobian or functional determinant JacyFð0;0Þ of Fwith respect to y at ð0;0Þ is the determinant of the complex ðp�pÞ�matrix whose elements are the (complex) partial derivativesF 0i;yj ð0;0Þ ð1r i; jrpÞ.

Let Rm denote the Euclidean space of dimension m andBðrÞ �Rm the open ball of center 0 and radius r40. The classicalimplicit function theorem for mappings of class C1 can be stated asfollows.

Theorem 2. If F : Bðr0Þ � BðR0Þ �Rn � Rp-Rp is of class C1 andsuch that

Fð0;0Þ ¼ 0; JacyFð0;0Þa0;

then there exist r1A ð0; r0Þ, R1Að0;R0Þ, and f : Bðr1Þ-BðR1Þ of classC1 such that, in Bðr1Þ � BðR1Þ,Fðx; yÞ ¼ 03y¼ f ðxÞ;

here the Jacobian JacyFð0;0Þ is the determinant of the real ðp�pÞ�matrix whose elements are the partial derivatives F 0i;yj ð0;0Þð1r i; jrpÞ. Of course, in Theorems 1 and 2, the centers 0 ofthe involved balls, chosen for simplicity, can be replaced by anyarbitrary point of the corresponding space. Those theoremsessentially give conditions under which a graph in the (x,y) spacedefined by a system of equations Fðx; yÞ ¼ 0 can be seen, in theneighborhood of one of its points ðx0; y0Þ as the graph of a functiony¼ f ðxÞ.

Although implicit functions were used much earlier, the com-plete statement and proof of Theorem 2 were only given by Dini(1878) in his mimeographed lectures of analysis of 1877–1878, andreproduced in the monographs of Angello Genocchi (written byGenocchi-Peano, 1884) and Jordan (1893).

In the special situation of Theorem 1 with p¼1, the followingresult gives information in cases where the Jacobian vanishes.F ðkÞy denotes the kth (complex) partial derivative with respect to y.

Theorem 3. If F : Bðr0Þ � BðR0Þ �Cn � C-C is analytic and suchthat

Fð0;0Þ ¼ F 0yð0;0Þ ¼⋯¼ Fðm�1Þy ð0;0Þ ¼ 0; F ðmÞ

y ð0;0Þa0; ð1Þ

then there exist r1A ð0; r0Þ, R1Að0;R0Þ, a0; a1;…; am�1 : Bðr1Þ-C

analytic, vanishing at 0, and G : Bðr1Þ � BðR1Þ-BðR1Þ analytic such

that Gðx; yÞa0 on Bðr1Þ � BðR1Þ and

Fðx; yÞ ¼ ½a0ðxÞþ⋯þam�1ðxÞym�1þym�Gðx; yÞ on Bðr1Þ � BðR1Þ:

In other words, the zeros of Fðx; �Þ in a neighborhood of ð0;0Þ arethe solutions of the algebraic equation

a0ðxÞþ⋯þam�1ðxÞym�1þym ¼ 0:

This result is usually called Weierstrass preparation theorem.For m¼1, it implies of course Theorem 1 with p¼1. As observed byLindelöf (1905), Cauchy stated and proved it already in 1831(Cauchy, 1831), and published it in 1841 (Cauchy, 1841). CarlWeierstrass stated and proved it in his Berlin's lectures around1860, and published it in 1886 (Weierstrass, 1886). Poincaré, as weshall see, stated, proved and published it in 1879.

3. 1879: Sur les propriétés des fonctions définies par leséquations aux différences partielles

Poincaré's (1879) thesis, entitled Sur les propriétés des fonctionsdéfinies par les équations aux différences partielles, and defended in1879, starts with some “preliminary lemmas”. The first one, calledby Poincaré “Théorème de Briot-Bouquet” is nothing but Theorem1. The unusual name given by Poincaré comes from the fact thatthe reference he gave for this theorem is the famous treatise onelliptic functions of Briot–Bouquet (1875). This may have pleasedBouquet, a member of the jury. Poincaré added that

this theorem can be seen as a consequence of the theorem ofexistence of the integral of a differential equation.1

Then Poincaré introduced the concept of an algebroïd function yfrom C to C, namely a function y which, in a neighborhood of0AC, is solution of an equation of the form

ymþAm�1ðxÞym�1þ⋯þA1ðxÞyþA0ðxÞ ¼ 0;

where mZ1 is an integer and the Aj vanish at 0 and are analyticnear 0.

If now Fðx; yÞ≔∑1k ¼ 0AkðxÞyk, where the analytic functions Aj of

x≔ðx1;…; xnÞACn are such that

A0ð0Þ ¼⋯¼ Am�1ð0Þ ¼ 0; Amð0Þa0

(which means that F is analytic in the neighborhood of ð0;0Þ andcondition (1) holds), Poincaré stated and proved the following tworesults as Lemmas 2 and 3.

Lemma 1. There exist m functions y(x) such that, near 0,

Fðx; yðxÞÞ ¼ 0 and limx-0

yðxÞ ¼ 0:

Lemma 2. The m functions y(x) are algebroïd of degree m.

So, a third independent author must be added to Cauchy andWeierstrass for essentially proving the preparation theorem.

In the thesis, those results are applied to the obtention of someextensions of Briot-Bouquet theorems for singular ordinary differ-ential equations to Cauchy–Kowalevski's problem for analyticpartial differential equations.

1 ce théorème peut être regardé comme une conséquence du théorème relatifà l'existence de l'intégrale d'une équation différentielle.

J. Mawhin / Studies in History and Philosophy of Modern Physics ∎ (∎∎∎∎) ∎∎∎–∎∎∎2





4. 1883–1884: Sur certaines solutions particulières duproblème des trois corps

Poincaré's first paper on the three body problem was publishedin 1884 (Poincaré, 1884) and announced in 1883 in a note to theComptes rendus (Poincaré, 1883b).

Poincaré considered the motion of three material points withrespective masses m;m0;M, in the situation where m=M and m0=Mare very small. He was interested by showing the existence ofperiodic solutions, i.e. solutions which return to the same position,with the same speed, after some period of time, in a suitablereference rotating system with respect to an inertial one. Further-more, those periodic solutions were requested to satisfy in addi-tion some symmetry properties that we do not describe here.

After a number of transformations, the problemwas reduced byPoincaré to finding the common zeros of three analytic functionsX;Y ; Z of the masses m;m0 (assumed to be small by taking M¼1)and of the six initial elements in a suitable system of astronomicalcoordinates. Poincaré wrote X ¼ Xðm;m0; x; y; zÞ in the form

X ¼ X0þX1þX2þ⋯þXnþ⋯; ð2Þwhere Xn is of the nth order in ðm;m0Þ, and similarly for Y ; Z.

Taking three of the initial elements constant, Poincaré assumedthat the three other ones ðx; y; zÞ could be chosen, after a transla-tion, such that

X0ð0;0;0;0;0Þ ¼ Y0ð0;0;0;0;0Þ ¼ Z0ð0;0;0;0;0Þ ¼ 0;

Jacðx;y;zÞðX0;Y0; Z0Þð0;0;0;0;0Þa0:

He then claimed that, if it is the case, for m and m0 small, nearbyinitial elements ðx; y; zÞ can be chosen so that,

Xðm;m0; x; y; zÞ ¼ Yðm;m0; x; y; zÞ ¼ Zðm;m0; x; y; zÞ ¼ 0;

i.e. so that ðx; y; zÞ are the initial conditions of periodic solutions ofthe three body problem.

The reader can immediately observe that this claim followsimmediately from Theorem 2, which was well known to Poincaré.Surprisingly he took another complicated way to prove it. Hestarted by observing that there exist small positive numbersa1; a2; a3 such that the points

ðX0ð0;0; x; y; zÞ; Y0ð0;0; x; y; zÞ; Z0ð0;0; x; y; zÞÞcover the parallelotope made of the ðX0;Y0; Z0Þ such that

X20oa21; Y2

0oa22; Z20oa23;

if one takes the initial elements ðx; y; zÞ in a suitable neighborhoodof ð0;0;0Þ. In modern terms, ðX0ð0;0; �Þ;Y0ð0;0; �Þ; Z0ð0;0; �ÞÞ is anopen mapping at 0. This is a consequence of the inverse mappingtheorem, itself equivalent to the implicit function theorem!

This allowed Poincaré to consider X;Y ; Z as functions of themasses m;m0 and of X0;Y0; Z0. Now, for m;m0 small with respect toa1; a2; a3; it follows immediately from (2) that

X40 for X0 ¼ a1; Xo0 for X0 ¼ �a1;

and similarly for Y ; Z. From this, Poincaré concluded that, forsufficiently small masses m;m0, some ðX0;Y0; Z0Þ (and hence someinitial elements ðx; y; zÞ) exist, such that X ¼ Y ¼ Z ¼ 0, by using thefollowing existence theorem.

Theorem 4. Let X1;X2;…;Xn be continuous functions of x1; x2;…; xnsuch that

Xi40 for xi ¼ ai; Xio0 for xi ¼ �ai ði¼ 1;…;nÞ:Then there exists x1; x2;…; xn such that jx1joa1; jx2joa2;…jxnjoanand

X1ðx1;…; xnÞ ¼ 0; X2ðx1;…; xnÞ ¼ 0;…; Xnðx1;…; xnÞ ¼ 0:

Geometrically, if, for each i¼ 1;…;n, the ith component ofX ¼ ðX1;…;XnÞ takes opposite signs on the opposite ith faces ofthe parallelotope, then X has a zero in the parallelotope. Thisstatement is clearly an n-dimensional generalization of Bolzano's(1817) intermediate value theorem for real continuous functions ofone real variable.

Poincaré (1884) gave the following heuristic proof when n¼2.For each fixed x1A ½�a1; a1�, the function X2 seen as a function of x2takes opposite signs at �a2 and a2 and hence vanishes between,giving a “curve” of zeros of X2 joining the opposite sides x1 ¼ �a1and x1 ¼ a1 of the rectangle. Similarly one obtains a “curve” ofzeros of X1 joining the opposite sides x2 ¼ �a2 and x2 ¼ a2 of therectangle. The two “curves” must meet in some point of therectangle, where X1 and X2 vanish simultaneously. To change thisargument into a real proof requires some algebraic topology!

Poincaré's rigorous proof of Theorem 4, identical in Poincaré(1883b, 1884), consists in the following three lines:

Mr. Kronecker has presented to the Berlin Academy, in 1869, aMemoir on functions of several variables; one can find there animportant theorem from which it is easy to deduce thefollowing result.2

Kronecker's (1869) memoir, 46 pages in two parts of hard analysis,contains many theorems. The one mentioned by Poincaré (wholeaves to his readers the task of finding it in Kronecker, 1869)implies the existence of a zero in an n-dimensional boundeddomain with smooth boundary, for a system of n smooth functionsof n variables defined there and having no common zero on theboundary, when the integral over the boundary of some expres-sion involving the Xi and their partial derivatives, is different fromzero. This integral is now referred as Kronecker integral orKronecker index.

Notice that Poincaré's Theorem 4 deals formally with contin-uous functions (which need not to be smooth) on a parallelotope(whose boundary is not smooth). Of course, “continuous” inPoincaré's language could mean “smooth”, and Kronecker's inte-gral makes sense on the boundary of a parallelotope. Poincaré's“easy deduction” must have been much less easy for his con-temporaries than for to-day's mathematicians familiar with topo-logical degree arguments and homotopy invariance!

For the experts, this Kronecker integral or Kronecker index, onthe smooth boundary ∂D of a bounded domain D�Rn, of thesmooth mapping X ¼ ðX1;…;XnÞ such that 0=2Xð∂DÞ is defined by

iK ½X; ∂D�≔1

μðSn�1Þ

Z∂DXns;

where the exterior differential ðn�1Þ�form s is defined by

s≔ ∑n

j ¼ 1ð�1Þj�1 yj

jyjn dy14⋯4dyj�14 cdyj 4dyjþ14⋯4dyn;

cdyj means that dyj is missing in the exterior product, μðSn�1Þ is themeasure of the ðn�1Þ�dimensional unit sphere, and n denotes thepull-back operation.

If X has only simple zeros x1;…; xm in D, Kronecker had shownthat

iK ½X; ∂D� ¼ ∑m

j ¼ 1sgn Jac XðxjÞ;

so that Kronecker index may be seen as an algebraic count of thenumber of zeros of X in D. The advantage of this algebraic countwith respect to the brute one is to remain constant for sufficiently

2 M. Kronecker a présenté à l'Académie de Berlin, en 1869, un Mémoire sur lesfonctions de plusieurs variables; on peut y trouver un important théorème d'où ilest aisé de déduire le résultat suivant.

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small perturbations of X. The extension of Kronecker's index tocontinuous X and arbitrary bounded open set D is the Brouwerdegree dB½X;D� (see e.g. Mawhin, 2004 for a definition andhistorical information).

Theorem 4, that Poincaré will not use any more in any furtherwork, is a global result, by contrast to the local character of theimplicit function theorem. Soon forgotten, it has been rediscov-ered by Cinquini (1940), and “proved” by an argument similar toPoincaré's heuristic one. One year later, Miranda (1941) showed inan elementary way its equivalence to Brouwer fixed point theorem,which asserts that any continuous transformation of a parallelo-tope into itself leaves at least one point fixed. Under the name ofMiranda theorem, Theorem 4 has been part of a ten years warbetween Cinquini and Giuseppe Scorza-Dragoni about its nature(topology or analysis?) and the validity of its various “proofs”.Poincaré's priority was only rediscovered in 1974, and the resultrenamed Poincaré–Miranda theorem. One can consult Mawhin(2007) for a detailed story and references, and Mawhin (2013)for an analytic proof independent of the concept of index ordegree.

Nowadays, the Poincaré–Miranda theorem, Brouwer fixedpoint theorem and degree arguments are widely used to obtainglobal results for the existence of periodic solutions and boundaryvalue problems associated to ordinary differential equations.

Incidently, Poincaré–Miranda's theorem allows one to prove anextension of Theorem 2 to the case where F is continuous on BðrÞ �BðRÞ and F 0y only exists at ð0;0Þ, but the uniqueness of f is lost. Theproof by induction of this result given in respected textbooks ofanalysis (for example de La Vallée Poussin, 1914; Young, 1910) arenot correct. More generally, Brouwer degree theory provides animplicit function theorem (again without uniqueness) for F con-tinuous only, when Fð0; yÞa0 on some sufficiently small sphere∂BðRÞ and the Brouwer degree dB½Fð0; �Þ;BðRÞ� is different from 0.The complex version of this generalization covers the assumptionsof Theorem 3, but does not provide the factorization.

As shown in Mawhin (2000), Kronecker index has been widelyused by Poincaré between 1881 and 1886. The first occurrence iscontained in his memoir of 1881 on the curves defined by adifferential equation (Poincaré, 1881). It is restricted to n¼2, anduses an equivalent definition given by Cauchy (1837) in 1837. OnFebruary 14, 1883, a letter of Kronecker (1869) called Poincaré'sattention to his memoir:

I would like to call your attention to a memoir that I havepublished in 1869 and that I take the liberty to send you […].I have developed there the generalization of this importanttheorem of Cauchy […] which seems to me to contain the truefoundations of function theory. It is very remarkable that thereexists a theorem completely analogous for an arbitrary numberof variables.3

It took a very short time to Poincaré to assimilate Kronecker'sdifficult memoir. In a paper submitted on July 20, 1883, Poincaré(1883a) applied Kronecker's index in dimension n to a study of thezeros of the functions Θ of several variables. Three months later,the note (Poincaré, 1883b) containing Theorem 4 and its applica-tion to the three body problem was presented to the Academy.Another application to the equilibrium of a rotating fluid body(Poincaré, 1885), discussed later, came out in 1885. A furtherapplication in 1886 to the qualitative theory of differential equa-tions of higher order (Poincaré, 1886) was the first one where

Poincaré cared to give the explicit definition of this Kroneckerintegral he had used so widely!

5. 1890: Sur le problème des trois corps et les équations de ladynamique

Implicit function arguments are often used in Poincaré'sfamous memoir of 1890 (Poincaré, 1890), crowned by King OscarII Prize (see Barrow-Green, 1997 for a detailed analysis of the twoversions of this work).

Theorem IV of Part 1, Chapter 1 is the analytic implicit functiontheorem, this time correctly attributed to Cauchy. When dealing inTheorem V with the preparation theorem, Poincaré wrote:

It would remain to examine what happens when the functionaldeterminant of the F with respect to the y is zero. This questionhas been the object of many researches, where one should firstquote the work of M. Puiseux on the roots of algebraicequations. I had myself the opportunity to consider analogousresearches in the first part of my thesis.4

Surprisingly, Poincaré omitted here to mention the importantcontributions of Weierstrass, Chairman of the jury of King OscarII prize!

Poincaré then showed that some cases of a null Jacobian inTheorem 1 could be reduced to Theorem 3 applied to a singleequation of the form Φðx; ynÞ ¼ 0, by successive elimination ofy1;…; yn�1.

In Chapter 2, Theorem 1 is used to prove that, given anautonomous differential system

_x ¼ Xðx; y; z;μÞ; _y ¼ Yðx; y; z;μÞ; _z ¼ Zðx; y; z;μÞ ð3Þdepending upon a small parameter μ, a contactless surface S, anda point P0 ¼ ðx0; y0; z0ÞAS close to A¼ ða0; b0; c0ÞAS, the nextintersection P1 ¼ ðx1; y1; z1Þ with S of the solution of (3) issuedfrom P0 (the first consequent) is a analytic mapping ofðx0�a0; y0�b0; z0�c0Þ and μ.

Chapter 3 is devoted to the study of periodic solutions ofdifferential systems of the form

_x ¼ Xðt; x;μÞ ð4Þwhere x¼ ðx1;…; xnÞ, X is T-periodic in t for some T40 (orindependent of t), and system (4) with μ¼ 0 has a T-periodicsolution φðtÞ. The question raised by Poincaré is the following one:

Under which conditions will we be allowed to conclude thatthe equations still possess periodic solutions for small valuesof μ?5

For jμj small, the solution of (4) equal to ϕð0Þþβ at t¼0 isdenoted by xðt;μ;βÞ. By uniqueness, xðt;0;0Þ ¼φðtÞ. Poincaréobserved that xðt;μ;βÞ will be T-periodic if and only if β is suchthat xðT ;μ;βÞ ¼φð0Þþβ, i.e. if and only if

ψ ðμ;βÞ≔xðT ;μ;βÞ�φð0Þ�β¼ 0:

Notice that ψ ð0;0Þ ¼ 0, and hence Theorem 2 implies that ifJacβψ ð0;0Þa0, then ψ ðμ;βÞ ¼ 0 has a solution β¼ βðμÞ for jμjsufficiently small, which proves the following result.

3 Je désirerais appeler votre attention à un mémoire que j'ai publié en 1869 etque je prend la liberté de vous envoyer. […] J'y ai développé la généralisation de cetimportant theéorème de Cauchy, qui me semble contenir le vrai fondement de lathéorie des fonctions. Il est très remarquable, qu'il existe un théorème tout-à-faitanalogue pour un nombre quelconque de variables.

4 Il nous resterait à examiner ce qui se passe quand le déterminant fonctionneldes F par rapport aux y est nul. Cette question a fait l'objet de recherchesnombreuses sur lesquelles je ne puis insister ici, mais au premier rang desquellesil convient de citer les travaux de M. Puiseux sur les racines des équationsalgébriques. J'ai eu moi-même l'occasion de m'occuper de recherches analoguesdans la première partie de ma Thèse inaugurale.

5 Dans quelles conditions aura-t-on encore le droit d'en conclure que leséquations comportent encore des solutions périodiques pour les petites valeursde μ?






Theorem 5. If Jacβψ ð0;0Þa0, system (4) has a T-periodic solutionfor all sufficiently small jμj.

In proving this result, Poincaré replaced here the use of hisn-dimensional intermediate value theorem by the classical impli-cit function theorem. This indirect disparition of Kronecker's namemay have pleased the Chairman of the jury of King Oscar II Prize.

Poincaré's method for finding periodic solutions of (4) that wehave just described can bring situations where the Jacobian of ψis zero. In some cases the elimination of β1;…;βn�1 as indicatedabove leads to an equation Φðμ;βnÞ ¼ 0, with β¼ 0 a root ofmultiplicity m of the equation Φð0;βÞ ¼ 0. Then (rememberingthat we are in the real case), Poincaré observed that a 2π�periodicsolution still exists form odd and jμj small, as a consequence of thepreparation theorem and the fact that a real algebraic equation ofodd order always has a real solution.

Situations where the Jacobian always vanish occur whensystem (4) has a first integral (for example Hamiltonian systems),or when system (4) with μ¼ 0 admits a family φðt;hÞ of T-periodicsolutions ðhARÞ. This is in particular the case when (4) does notdepend explicitly upon t. Then, the period is not known a prioriand will in general depend upon μ. Thus, for xðt;μ;βÞðTþτÞ�periodic, ψ1;…;ψn will depend on the nþ1 variablesβ1;…;βn; τ. But xðtþh;μ;βÞ will also be ðTþτÞ�periodic for allhAR, 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.

6. 1892–1899: Les méthodes nouvelles de la mécanique céleste

This fundamental book (Poincaré, 1892), which developed thecrowned memoir (Poincaré, 1890) into 3 volumes totalizing morethan 1250 pages, contains many applications of implicit functiontechniques.

Chapter 2 of Volume 1 essentially repeats the standard resultson implicit functions, the preparation and elimination theoremsgiven in Poincaré (1890). More surprisingly, Kronecker's indexreappears in the last section of this chapter to show that if a realsmooth function FðxÞ≔Fðx1;…; xnÞ has a strict maximum at 0, sothat

F 0ð0Þ≔ðF 0x1 ð0Þ;…; F 0xn ð0ÞÞ ¼ 0;

then 0 is a solution of odd order of F 0ðxÞ ¼ 0:To prove this result, Poincaré considered the small closed level

surface S of equation FðxÞ ¼ Fð0Þ�λ2 and a family of functionsΦðx;μÞ such that Φðx;0Þ ¼ FðxÞ, and the zeros ξ1;…; ξm of Φ0

xðx;μÞare simple. Therefore, by the properties of Kronecker index,

iK ½F 0; S� ¼ 1¼ iK ½Φ0xð�;μÞ; S�

for jμj sufficiently small, and

1¼ fξj : Jacx Φ40g�#fξj : Jacx Φo0g

m¼ fξj : Jacx Φ40gþfξj : Jacx Φo0g;where #E denotes the number of elements of the set E. Conse-quently,

1þm¼ 2#fξj : Jacx Φ40g;and m is odd.

Chapter 3 of Volume 3 is devoted to periodic solutions ofsystems depending upon a small parameter. The general theory isclose to that given in Poincaré (1890). Its application to the threebody problem develops the results of Poincaré (1883b), usingimplicit function techniques instead of Theorem 4.

Some special attention is paid to Hamiltonian systems depend-ing upon a parameter, when the unperturbed system has a zeroHessian. The study of non-trivial periodic solutions near an

equilibrium 0 of systems of the form

_x ¼ Xðx;μÞsuch that Xð0;μÞ ¼ 0 for all μ again leads to an implicit functionproblem with zero Jacobian, and is briefly considered.

Those results of Poincaré have inspired further work of Bliss(Bliss, 1913), William D. MacMillan (MacMillan, 1910a, 1910b,1912a, 1912b, 1912c, 1936), Forest R. Moulton (Moulton, 1912,1920, 1930), Ioel G. Malkin, Earl A. Coddington, Norman Levinson,Warren S. Loud, many others (see references in Cesari, 1963;Verhulst, 2012).

Chapter 4 is devoted to the theory of characteristic exponents.Considering again system (4) which, for μ¼ 0, has a T-periodicsolution φðtÞ, the variational equation (already considered by Jacobiand Darboux in other contexts)

_y ¼ X0xðt;φðtÞ;0Þy ð5Þ

is introduced. As shown by Floquet (1883), the fundamentalmatrix solution Y(t) of (5) has the form YðtÞ ¼ ZðtÞetS, where Z(t)is T-periodic and S is a constant matrix, whose eigenvalues are thecharacteristic exponents of (5). A zero characteristic exponentcorresponds to a T-periodic solution of (5). Poincaré then provedthe following existence result.

Theorem 6. If all characteristic exponents of (5) are different fromzero, system (4) has a T-periodic solution for small jμj.

In the case of an autonomous system

_x ¼ Xðx;μÞ ð6Þhaving, for μ¼ 0 a T-periodic solution φðtÞ, the correspondingvariational equation

_y ¼ X0xðφðtÞ;0Þy ð7Þ

always has the T-periodic solution _φðtÞ, and hence a characteristicexponent equal to 0. Poincaré proved for (6) the following variant ofTheorem 6.

Theorem 7. If n�1 characteristic exponents of (7) are different fromzero, system (6) has a periodic solution of period close to T for smalljμj.

The proofs of Theorems 6 and 7 are again based upon implicitfunction techniques.

Theorem 6 can be seen as an anticipation of an implicitfunction theorem in the frame of infinite dimensional spaces ofT-periodic functions. Indeed, if we define the mapping F fromC1T � R into CT, where C1

T is the space of T-periodic functions ofclass C1, and CT the space of continuous T-periodic functions, bythe formula

F ðx;μÞ≔ _xð�Þ�Xð�; xð�Þ;μÞ;then the Fréchet derivative F 0

xðφ;0Þ of F at ðφ;0Þ is the linearoperator from C1T into CT given by

F 0xðφ;0Þyð�Þ ¼ _yð�Þ�X0

xð�;φð�Þ;0Þyð�Þ;i.e. the linear operator between C1T and CT associated to thevariational equation. The conditions on the characteristic expo-nents in Theorem 6 insure the invertibility of this Fréchetderivative, corresponding the nonvanishing of the Jacobian in thefinite-dimensional setting. Of course, none of those functionalanalytic concepts were defined when Poincaré published his book(Poincaré, 1892).

Implicit function theorems are still used in Chapter 5 devotedto the problem of non-existence of other uniform integrals that theclassical ones in the three body problem, in Chapters 28, 30 and 31to study the periodic solutions of second kind (genre) (i.e. ofsmallest period kT for some integer k41), their formation andtheir properties. The reader is invited to consult (Verhulst, 2012)






for a detailed analysis of the content of Poincaré (1892) and of itsmodern consequences.

7. 1895: Sur l’équilibre d'une masse fluide animée d'unmouvement de rotation

Starting with the ellipsoid of revolution, new figures of equili-brium of rotating fluid bodies submitted to gravitation have beenobtained, after Isaac Newton, by Colin MacLaurin, Carl-GustafJacobi, Bernhard Riemann, Joseph Liouville and others (see thehistorical information in Appell, 1921).

Poincaré (1885) had the idea of considering the evolution of thevarious known equilibria and possible new (pear-shaped) ones, byintroducing the concept of series of equilibria with the angularvelocity as parameter. He wanted to show that new shapes ofequilibrium could bifurcate from another one for some values ofthe parameter, and exchange their stability. Such results were ofcourse important in the discussion of the evolution of planets andcelestial bodies.

To motivate his results, Poincaré started with two sections (IIand III) devoted to the simpler situation of the equilibria of asmooth potential

Fðx; y1; y2;…; ynÞdepending on a parameter x and on a finite number of variablesy1;…; yn. Those equilibria satisfy the equations

F 0y1 ðx; yÞ ¼ F 0y2 ðx; yÞ ¼⋯¼ F 0yn ðx; yÞ ¼ 0;

and it is assumed that 0 is an equilibrium when x¼0, namely that

F 0y1 ð0;0Þ ¼ F 0y2 ð0;0Þ ¼⋯¼ F 0yn ð0;0Þ ¼ 0:

If the Hessian detðF″yj ;yk ð0;0ÞÞa0, the implicit function theoremimplies the existence of a locally unique solution, i.e. of a uniquebranch of equilibria passing through ð0;0Þ and close to this point.Hence a necessary condition for bifurcation, i.e. for another branchto arise at ð0;0Þ, is that

detðF 0 0yj ;yk ð0;0ÞÞ ¼ 0: ð8Þ

So, in this setting, bifurcation is the study of what happens whenthe conditions of the implicit function theorem are not satisfied.

In the special case where n¼1 and F 0yð0;0Þ ¼ 0, Poincaréshowed by an elementary reasoning that if y¼ f ðxÞ such thatf ð0Þ ¼ 0 satisfies the equation F 0yðx; f ðxÞÞ ¼ 0 near 0, a sufficientcondition for the existence of another branch y¼ gðxÞ such thatgð0Þ ¼ 0, i.e. a sufficient condition for bifurcation at ð0;0Þ is that thefunction ΔðxÞ≔F″y;yðx; f ðxÞÞ changes sign at 0.

The situation is more complicated when n¼2. The equilibriumcondition at 0 gives the system

F 0y1 ð0;0;0Þ ¼ F 0y2 ð0;0;0Þ ¼ 0:

Let

F 0yðx; y1; y2Þ≔ðF 0y1 ðx; y1; y2Þ; F0y2ðx; y1; y2ÞÞ:

For x fixed, let iK ½F 0yðx; �; �Þ;C� denote the Kronecker index of F 0yðx; �Þalong a small circle C centered at ð0;0Þ. Using once more theproperties of Kronecker index, Poincaré has proved the followingsufficient condition for bifurcation.

Theorem 8. If iK ½F 0yð�ɛ; �; �Þ;C�a iK ½F 0yðþɛ; �; �Þ;C� for all sufficientlysmall ɛ40, then ð0;0;0Þ is a bifurcation point of F 0y.

A closer look to Poincaré's proof shows that it remains valid forarbitrary n and when the gradient mapping F 0y is replaced by ageneral smooth one. This result of Poincaré clearly anticipates themodern topological approach for bifurcation theory. One can

consult Mawhin (2011) for details and references to the contem-porary literature.

As a special case of Theorem 8, Poincaré showed that if y¼ f ðxÞis such that f ð0Þ ¼ 0 satisfies the equations

F 0y1 ðx; f ðxÞÞ ¼ F 0y2 ðx; f ðxÞÞ ¼⋯¼ F 0yn ðx; f ðxÞÞ ¼ 0 ð9Þ

near 0 and if ΔðxÞ≔detðF″yj ;yk ðx; f ðxÞÞÞ changes sign at x¼0, thenð0;0Þ is a bifurcation point of (9), which generalizes the elementaryresult obtained for n¼1.

Poincaré then considered the quadratic form

ΦðX1;X2;…;XnÞ≔ ∑n

i;k ¼ 1F″yi ;ykXiXk;

whose coefficients are called coefficients of stability and whosediscriminant along a branch of equilibria is ΔðxÞ, to study theexchange of stability of the equilibria in the branches at abifurcation point. A linear change of variable allows one to cancelall the F″yi ;yk with iak, so that the new stability coefficients areF″y1 ;y1 ;…; F″yn ;yn . One of them must vanish at 0, and if only one, sayF″y1 ;y1 does so, Theorem 2 applied to F 0y2 ¼ 0 provides y2 as afunction of x; y1; y3;…; ynÞ. Introducing this function in F 0y3 ¼ 0and using again Theorem 2, one obtains y3 as a function ofðx; y1; y4;…; ynÞ. Continuing in this way leads to a single equationof the form

Gðx; y1Þ≔ay21þ2bxy1þcx2þhigher order terms¼ 0;

which gives near the origin two real branches

y1 ¼ψ1ðxÞ; y1 ¼ψ2ðxÞif b2�ac40. Poincaré's analysis of the sign of Δ along thosebranches shows that there is exchange of stability at their meetingpoint ð0;0Þ.

The next section of Poincaré (1885), devoted the case of aninfinite number of variables, starts as follows:

The problems treated in the two previous sections do notpresent any type of difficulty. Unfortunately, when onesearches the figure of equilibrium of a fluid mass submittedto various forces, the question is much more complicated.Indeed, the figure of such a mass depends, not on a finitenumber of variables y1; y2;…; yn, but on a infinite number ofvariables.6

The reader can make his own opinion about the total absence ofdifficulty in the previous analysis.

8. 1902: Figures d’équilibre d'une masse fluide

Poincaré returned to the problem of the figures of equilibriumof rotating fluid bodies in his lectures at the Sorbonne of 1900,written down by L. Dreyfus and published it in 1902 (Poincaré,1902).

The second section “Stabilité des figures trouvées” of Chapter VII“Attraction des ellipsoïdes” starts by considering, up to a change ofnotations, the question treated in Sections II and III of the paper(Poincaré, 1885), namely the bifurcation of branches of equilibriumof a potential F depending upon one parameter and a finite numberof variables. The necessary condition (8) for bifurcation is obtainedin a similar way.

6 Les problèmes traités dans les deux paragraphes précédents ne présententaucune espèce de difficulté. Malheureusement, lorsqu'on recherche la figured’équilibre d'une masse fluide soumise à diverses forces, la question est beaucoupplus compliquée. En effet, la figure d'une pareille masse dépend, non pas d'unnombre fini de variables y1 ; y2 ;…; yn , mais d'un nombre infini de variables.






Using this time implicit function techniques only, and underthe assumption that one of the minors of detðF″yi ;yk ð0;0ÞÞ at least isdifferent from zero, successive reductions based upon Theorem 2allow Poincaré to write yj ¼φjðy1; xÞ for some function φj

ð2r jrnÞ. Lettingψ ðy1; xÞ≔Fðy1;φ2ðy1; xÞ;…;φnðy1; xÞ; xÞ;the equilibrium condition becomes ψ 0

y1ð0;0Þ ¼ 0. For this equation,

a necessary condition for bifurcation is that ψ ″y1 ;y1

ð0;0Þ ¼ 0. Theshape of the bifurcation is discussed by Poincaré in terms of ψ ″

y1 ;x,

and the zones of stability of the equilibrium are described in theplane ðy1; xÞ. In contrast to Poincaré (1885), a number of picturesillustrate the reasonings and the results.

Poincaré's discussion is reproduced with more details in thelast chapter “Étude de la stabilité des figures d’équilibre” of thebook (Appell, 1921), and illustrated by the interesting example of amaterial point moving with friction on a rotating circle.

9. Conclusions

A general conclusion of this analysis of the role and use ofimplicit function techniques in Poincaré's work on the qualitativetheory of differential equations, the three body problem and thefigures of equilibrium of rotating fluid bodies is that the paths of agenius are unpredictable. Clearly, Poincaré knew very well theimplicit function theorem before writing his thesis, and he evenrediscovered the preparation theorem independently of Cauchyand Weierstrass.

However, in his first papers on the periodic solutions of thethree body problem and on the figures of equilibrium of rotatingfluid bodies, Poincaré used efficiently Kronecker index (at thistime a sophisticated and little known tool) in situations where theimplicit function theorem was sufficient, as confirmed by hisfurther contributions in papers or monographs, where Kroneckerindex has been abandoned in favor of more classical techniques.

But, this superfluous detour gave him the opportunity to createthe topological approach to nonlinear differential equations andbifurcation theory, from which not only local results, but also andmainly global results, outside of the scope of implicit functiontheorems, can be obtained.

The paths of the Lord are claimed to be unpredictable. WithPoincaré, it is not only often difficult to understand “how he did itthis way”, but still more difficult to know “why he did it that way”.

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The implicit function theorem and its substitutes in Poincaré׳s qualitative theory of differential equations