Poincar e and the Three-Body Problem - Bourbaphy

1 Poincar e, 1912-2012, S eminaire Poincar e XVI (2012) 45 133S eminaire Poincar ePoincar e and the Three-Body ProblemAlainChencinerObservatoire de Paris, IMCCE (UMR 8028), ASD77, avenue Denfert-Rochereau, 75014 Paris, D epartement de math ematique, Universit e Paris Three-Body Problem has been a recurrent theme of Poincar e sthought. Having understood very early the need for a qualitative study of non-integrable differential equations, he developed the necessary fundamental tools:analysis, of course, but also topology, geometry, probability. One century later,mathematicians working on the Three-Body Problem still draw inspiration fromhis works, in particular in the three volumes ofLes m ethodes nouvelles de lam ecanique c elestepublished respectively in 1892, 1893, Introduction452 General Problem of Equations of theN- body Problem .. Kepler Problem : Delaunay and Poincar e coordinates.

2 Planetary Problem : heliocentric coordinates .. Reduced Problem .. General Problem of dynamics .. The basic intuition .. 553 Next approximation: Lagrange s and Laplace s secular Kepler degeneracy: fast and slow variables .. Averaged system .. Quadratic part and singularities .. 584 Periodic solutions 1) Local existence by P en etrer dans une place jusqu ici r eput ee inabordable .. The three sorts .. and some others .. 655 Quasi-periodic solutions 1) Formal aspects: Lindstedt What is a Lindstedt series? .. Different types of Lindstedt series .. From the old methods to the new ones and back .. One torus or a foliation by tori? ( Poincar e s ambiguities) .. Existence of the Lindstedt series in the non-degenerate case .. Coping with the.

3 7546A. ChencinerS eminaire Poincar forgetting a resonance! .. 776 Periodic solutions 2) The source of Variational equations: exponents, stability, asymptotic solutions .. Non-existence of uniform first integrals .. Divergence of Lindstedt series .. Back to the old methods: the Lessons of Celestial Mechanics .. Complexifying Poincar e s method .. 867 Resonances 1) Bohlin Circulation, libration, separatrix: what is a Bohlin series? .. Bohlin method in the non-degenerate case .. 888 Integral invariants and Poisson Integral invariants as integrals of the variational equations .. Poisson stability .. From the recurrence theorem to ergodic theory .. 969 Stroboscopy 1) Planar Circular Restricted Three-Body The simplest Three-Body Problem .. Hill stability .. Return map.

4 10010 Resonances 2) Homoclinic and heteroclinic Divergence of the Bohlin series and exponentially small splitting ..cette figure, que je ne cherche m eme pas `a tracer .. 10411 Quasi-periodic solutions 2) Analytic aspects: New methods, chapter XIII, section 149 .. Arnold s theorem for the planar Problem .. Dealing with the spatial Problem .. 11012 Stroboscopy 2) What we understand of the dynamics of the return map Bifurcations: subharmonics .. The last geometric theorem and twist diffeomorphisms of the annulus Computer experiments and the geometrization of phase space .. 11613 A great principle of physics and some Least action and instability .. Minimizing the action: the note of 1896 .. Solutions of the second species .. 12014 Resonances 3) Arnold diffusion .. The oldest open question in dynamical systems .. 12115 Surprises of a eulogy12316 A seminar123 Poincar e, 1912-2012, Vol.

5 XVI, 2012 Poincar e and the Three-Body Problem4717 Thanks12318 Regret12419 Note on the references1241 IntroductionSince the time of Newton himself, the Three-Body Problem was a major sourceof development of analysis: it is enough to mention the names of Euler, Clairaut,d Alembert, Laplace, Lagrange, Jacobi, Cauchy,..At the end of the nineteenthcentury, Poincar e opened a new era, introducing geometric, topological and prob-abilistic methods in order to understand qualitatively the incredibly complicatedbehavior of most of the solutions of this Problem . At the same time, he analysed themethods used by the astronomers in order to understand the short-term motionsusing divergent series and, as emphasized in a recent paper by Ramis [Ra1], heforesaw many aspects for the present development of this theme. From 1883, that isonly 4 years after his thesis defense, and until his death in 1912, Poincar e publishedmajor papers on (or motivated by) the Three-Body Problem .

6 Already in 1881, inthe introduction to the first part of hisM emoire sur les courbes d efinies par une equation diff erentielle[P3], he took it as motivation for a qualitative global study:Prenons, par exemple, le probl`eme des trois corps : ne peut-on pas sedemander si l un des corps restera toujours dans une certaine r egion duciel ou bien s il pourra s eloigner ind efiniment ; si la distance de deux corpsaugmentera, ou diminuera `a l infini, ou bien si elle restera comprise entrecertaines limites ? Ne peut-on pas se poser mille questions de ce genre,qui seront toutes r esolues quand on saura construire qualitativement lestrajectoires des trois corps ? Et, si l on consid`ere un nombre plus grand decorps, qu est-ce que la question de l invariabilit e des el ements des plan`etes,sinon une v eritable question de g eom etrie qualitative, puisque, faire voirque le grand axe n a pas de variations s eculaires, c est montrer qu il oscilleconstamment entre certaines limites.

7 Tel est le vaste champ de d ecouvertesqui s ouvre devant les g eom` 1885, introducing the third part of the sameM emoire, he was more precise andaddressed the reader in his characteristic style:On n a pu lire les deux premi`eres parties de ce M emoire sans etre frapp ede la ressemblance que pr esentent les diverses questions qui y sont trait eesavec le grand probl`eme astronomique de la stabilit e du syst`eme solaire. Cedernier probl`eme est, bien entendu, beaucoup plus compliqu e, puisque les equations diff erentielles du mouvement des corps c elestes sont d ordre tr`es1 Let us take, for example, the Three-Body Problem : is it not possible to ask whether one of the bodies will remainforever in some region of the sky or whether it will possibly get away indefinitely; whether the distance beween twobodies will increase, or decrease indefinitely, or whether it will stay bounded between some limits?

8 Is it not possibleto ask a thousand similar questions, which will all be solved as soon as one is able to construct qualitatively thetrajectories of the three bodies? And if one considers more bodies, what is the question of the invariability of theelements of the planets but a true question of qualitative geometry, as showing that the great axis has no secularvariations amounts to showing that it oscillates between some limits. Such is the vast field of discoveries which opensup to ChencinerS eminaire Poincar e elev e. Il y a m eme plus, on rencontrera, dans ce probl`eme, une difficult enouvelle, essentiellement diff erente de celles que nous avons eu `a surmonterdans l etude du premier ordre, et j ai l intention de la faire ressortir, sinondans cette troisi`eme Partie, du moins dans la suite de ce difficulty alluded to by Poincar e is caused by the so-calledsmall denomina-torswhich appear in the perturbation series of the astronomers.

9 And indeed, havingdevoted the end of theM emoireto the study of theLindstedt serieswhich governthe problematic existence of families of quasi-periodic solutions3 surrounding aperiodic one inR3, he concluded as follows:D apr`es ce qui pr ec`ede, on comprendra sans peine `a quel point les difficult esque l on rencontre en M ecanique c eleste, par suite des petits diviseurs et dela quasi-commensurabilit e des moyens mouvements, tiennent `a la naturem eme des choses et ne peuvent etre tourn ees. Il est extr emement probablequ on les retrouvera, quelle que soit la m ethode que l on landmark in Poincar e s works on the Three-Body Problem is the famous MemoirSur le probl`eme des trois corps et les equations de la dynamique[P1], winner in 1889of the prize given on the occasion of the 60th birthday of the King of Sweden, andeven moreLes m ethodes nouvelles de la m ecanique c eleste[P2] whose three volumes,totaling almost 1300 pages, appear respectively in 1892, 1893 and 1899.

10 Vastly en-larging the scope of the Memoir, this extraordinary work, which encompasses anddevelops most of Poincar e s previous researches on the Three-Body Problem , is thesource of a major part of the modern theory ofDynamical Systems:normal forms,exponents, invariant manifolds, homoclinic and heteroclinic solutions, analytic non-integrability, divergence of the perturbation series and exponentially small splitting ofseparatrices, variational equations and integral invariants, generating functions, re-currence theorem, surfaces of section and return maps, twisting property, all of themare part of the present landscape and they paved the way forbifurcation studies andthe theory of singularities, symbolic dynamics, invariant measures and ergodic the-ory, , weak and diffusion, symplectic also a wealthof computer experiments. Indeed, in the realm of Dynamical Systems, new ideaswhich are not in one way or another rooted in Poincar e s works are few and farbetween ([AKN, HK, C0]).