Henri Poincaré : a biography through the daily papers

261 Pages·2014·6.475 MB·English

by  Gérini, Christian, Ginoux, Jean-Marc, Poincaré, Henri

HENRI POINCARÉ
A Biography Through the Daily Papers
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HENRI POINCARÉ
A Biography Through the Daily Papers
Jean-Marc Ginoux
LSIS, CNRS, Université de Toulon, France
Archives Henri Poincaré, CNRS, Université de Nancy, France
Christian Gerini
Université du Sud Toulon Var, France
& Université Paris-11 Orsay, France
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HENRI POINCARÉ
A Biography Through the Daily Papers
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“Thought is only a flash between two long nights, but this flash
is everything.”
— H.Poincar´e1
1The Value of Science.
v
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Foreword
Henri Poincar´eis known all over the worldas a famous astronomer, math-
ematician, physicist and philosopher. He has been dubbed by Eric Temple
Bellasthe“LastUniversalist”. Asamathematicianandphysicist,hemade
manyoriginalfundamentalcontributionstopureandappliedmathematics,
mathematical physics, and celestial mechanics.
In his seminal works “On the curves defined by differential equa-
tions” Poincar´e (1881-1886) was led to define what is nowadays called a
“dynamical system” to model the evolution of phenomena by using ordi-
nary differential equations (O.D.E.). Then, he developed many tools to
analyze the behavior (stability) of the solutions of such systems including
“characteristic exponents” and “first return maps” which has since been
named in his honor as “Poincar´e maps”. In 1889, Poincar´e won the Prize
Competition sponsored by the king Oscar II of Sweden and Norway prov-
ing thus the efficiency of his new mathematical concepts to establish the
stability of the Solar system. Unfortunately, very few months later, the
Swedish mathematician Lars Edvard Phragmen responsible for proofread-
ing, discovered a mistake in the memoir (entitled: “On the three body
problem and the problem of dynamics”) sent by Poincar´e. The Editor in
Chief, G¨ostaMittag-LefflerinformedPoincar´eofthis probleminJuly 1889
and here is what he answered in December 1889:
“Ithoughtthatalltheseasymptoticcurves(homoclinicorbits)
after moving away from a closed curve representing a periodic
solution would then asymptotically approach the same closed
curve. What is true is that there is an infinity of them that
enjoy this property. I will not conceal from you the distress
that this discovery has caused me.”
vii
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viii Henri Poincar´e: a biography through the daily papers
Ithassincebeennoticed(seetheworksofJ.-C.Yoccoz)thatthis“fruit-
fulmistake”ledPoincar´etothe discoveryofdeterministic chaos. Hisletter
to Mittag-Leffler ends with this sentence:
“I will write more when I see a little clearer in my works.”
It will take ten years for Poincar´e to “see a little clearer” in his works.
Ten years during which he wrote his famous “New Methods of Celestial
Mechanics”(1892-1899). Inthethirdvolume,Poincar´e(1899)wroteabout
the homoclinic orbits which nowadays is considered as the signature of
chaos:
“The complexity of this figure will be striking, and I shall not
even try to draw it.”
Thus, as briefly recalled above, Poincar´e’s contribution to Dynamical
Systems Theory and more particularly to Chaos Theory is obviously of
great importance. In my own works concerning Nonlinear Circuit The-
oryandCellularAutomata, Poincar´e’smethodswerethefoundationsupon
whichsomeofmydemonstrationsarebasedon. Asanexample,inanarticle
entitled: The Double Scroll family (Chua et al., IEEE, 11, pp. 1073-1118,
1986), I used the Poincar´e and half-return maps to establish a Rigorous
Proof of Chaos in my electronic circuit (Chua’s circuit). Although the im-
portance ofPoincar´e’sworksis universallyrecognizedin Pure andApplied
Mathematics, in Physics, in Astronomy and in Philosophy, and in spite of
the factthat many workshavebeen devotedto them, veryfew biographies
have been written about his life. In the middle of the thirties Eric Temple
Bell (1937) included in his book Men of Mathematics a chapter entitled
“Henri Poincar´e: the last universalist” in which he told many anecdotes
about him. Here is one:
“One phase of Poincar´e’s absent-mindedness resembles some-
thing quite different. Thus (Darboux does not tell the story,
but it should be told, as it illustrates a certain brusqueness
ofPoincar´e’s lateryears),when adistinguished mathematician
had come all the way from Finland to Paris to confer with
Poincar´eonscientificmatters,Poincar´edidnotleavehisstudy
to greet his caller when the maid notified him, but continued
to pace back and forth – as was his custom when mathemati-
cizing – for three solid hours. All this time the diffident caller
sat quietly intheadjoining room, barred from themasteronly
by flimsy porti`eres. At last the drapes parted and Poincar´e’s
buffaloheadwasthrustforaninstantintotheroom. “Vousme
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Foreword ix
derangezbeaucoup”(Youaredisturbingmegreatly), thehead
exploded,anddisappeared. Thecallerdepartedwithoutanin-
terview,whichwasexactlywhatthe“absent-minded”professor
wanted.” (Bell, 1937, p. 588)
Regardinghisworkingmethod,Bellsaid,quotingGastonDarbouxwho
was the President of the jury of the thesis:
“Poincar´ewasanintuitionist. Havingoncearrivedatthesum-
mitheneverretracedhissteps. Hewassatisfiedtohavecrashed
throughthedifficulties andleft toothersthepainsof mapping
theroyalroadsdestined tolead moreeasily totheend.” (Bell,
1937, p. 592)
TheaboveanecdotesprovidesomeinsightonPoincar´e’spersonalityand
allow us to better understand his work and his thought. Indeed, the work
of a genius is an inexhaustible source of knowledge in which we should go
over with a fine-toothed comb in order to draw the ideas that lead to new
theories.
Leon Chua
University of California, Berkeley
Technische Universita¨t, Mu¨nchen