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Robert HookeBorn: 18-Jul-1635Birthplace: Freshwater, Isle of Wight, EnglandDied: 3-Mar-1703Location of death: London, EnglandCause of death: unspecified

Gender: MaleRace or Ethnicity: WhiteSexual orientation: StraightOccupation: Scientist

Nationality: EnglandExecutive summary: Law of Elasticity, microscopist

English experimental philosopher, born on the 18th of July 1635 at Freshwater,in the Isle of Wight, where his father, John Hooke, was minister of the parish.After working for a short time with Sir Peter Lely, he went to WestminsterSchool; and in 1653 he entered Christ Church, Oxford, as servitor. After 1655he was employed and patronized Robert Boyle, who turned his skill to accountin the construction of his air-pump. On the 12th of November 1662 he was appointed curator of experiments to the RoyalSociety, of which he was elected a fellow in 1663, and filled the office during the remainder of his life. In 1664 Sir JohnCutler instituted for his benefit a mechanical lectureship of £50 annually, and in the following year he was nominatedprofessor of geometry in Gresham College, where he subsequently resided. After the Great Fire of 1666 he constructed amodel for the rebuilding of the city, which was highly approved, although the design of Sir Christopher Wren waspreferred. During the progress of the works, however, he acted as surveyor, and accumulated in that lucrative employment asum of several thousand pounds, discovered after his death in an old iron chest, which had evidently lain unopened forabove thirty years.

Hooke fulfilled the duties of secretary to the Royal Society during five years after the death of Henry Oldenburg in 1677,publishing in 1681-82 the papers read before that body under the title of Philosophical Collections. A protractedcontroversy with Johannes Hevelius, in which he urged the advantages of telescopic over plain sights, brought him little butdiscredit. His reasons were good; but his offensive style of argument rendered them unpalatable and himself unpopular.Many circumstances concurred to embitter the latter years of his life. The death, in 1687, of his niece, Mrs. Grace Hooke,who had lived with him for many years, caused him deep affliction; a lawsuit with Sir John Cutler about his salary (decided,however, in his favor in 1696) occasioned him prolonged anxiety; and the repeated anticipation of his discoveries inspiredhim with a morbid jealousy. Marks of public respect were not indeed wanting to him. A degree of M.D. was conferred onhim at Doctors' Commons in 1691, and the Royal Society made him, in 1696, a grant to enable him to complete hisphilosophical inventions. While engaged on this task he died, worn out with disease, on the 3rd of March 1703 in London,and was buried in St. Helen's Church, Bishiopsgate Street.

In personal appearance Hooke made but a sorry show. His figure was crooked, his limbs shrunken; his hair hung indishevelled locks over his haggard countenance. His temper was irritable, his habits penurious and solitary. He was,however, blameless in morals and reverent in religion. His scientific achievements would probably have been more strikingif they had been less varied. He originated much, but perfected little. His optical investigations led him to adopt in animperfect form the undulatory theory of light, to anticipate the doctrine of interference, and to observe, independently ofthough subsequently to F. M. Grimaldi (1618-1663), the phenomenon of diffraction. He was the first to state clearly that themotions of the heavenly bodies must be regarded as a mechanical problem, and he approached in a remarkable manner thediscovery of universal gravitation. He invented the wheel barometer, discussed the application of barometric indications tometeorological forecasting, suggested a system of optical telegraphy, anticipated E.F.F. Chladni's experiment of strewing avibrating bell with flour, investigated the nature of sound and the function of the air in respiration and combustion, andoriginated the idea of using the pendulum as a measure of gravity. He is credited with the invention of the anchorescapement for clocks, and also with the application of spiral springs to the balances of watches, together with theexplanation of their action by the principle Ut tensio sic vis (1676).

His principal writings are Micrographia (1664); Lectiones Cutlerianae (1674-79); and Posthumous Works, containing asketch of his "Philosophical Algebra", published by R. Waller in 1705.

Father: John Hooke (minister)

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Une autre solution d'échange

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deja contenue dans un pli cachete depose a la fin de 1747 et aujourd’hui perdu ; il n’auraitdonc manque a d’Alembert que d’appliquer numeriquement sa methode pour faire la memedecouverte que Clairaut. Il en retire une amertume qui s’exprimera dix ans plus tard dansle tome 9 de l’Encyclopedie (le volume paraıt en 1765 mais l’article est redige en 1759) : “Jene dois pas oublier d’ajouter 1o. que ma methode pour determiner le mouvement de l’apogee,est tres-elegante & tres-simple, n’ayant besoin d’aucune integration, & ne demandant quela simple inspection des coefficients du second terme de l’equation differentielle ; 2o. quej’ai demontre le premier par une methode rigoureuse, ce que personne n’avoit encore fait,& n’a meme fait jusqu’ici, que l’equation de l’orbite lunaire ne devoit point contenir d’arcsde cercle ; si on ajoute a cela la maniere simple & facile dont je parviens a l’equationdifferentielle de l’orbite lunaire, sans avoir besoin pour cela, comme d’autres geometres,de transformations & d’integrations multipliees ; & le detail que j’ai donne ci-dessus demes travaux & de ceux des autres geometres, on conviendra, ce me semble, que j’ai eu plusde part a la theorie de la lune que certains mathematiciens n’avoient voulu le faire croire.Je ne dois pas non plus passer sous silence la maniere elegante dont M. Euler integrel’equation de l’orbite lunaire ; methode plus simple & plus facile que celle de M. Clairaut& que la mienne ; & cette observation jointe a ce que j’ai dit plus haut des travaux dece grand geometre, par rapport a la lune, suffira pour faire voir qu’il a aussi travaille tresutilement a cette theorie, quoiqu’on ait aussi cherche a le mettre a l’ecart autant qu’on apu. L’Encyclopedie faite pour transmettre a la posterite l’histoire des decouvertes de notresiecle, doit par cette raison rendre justice a tout le monde ; & c’est ce que nous croyonsavoir fait dans cet article. Comme ce manuscrit est pret a sortir de nos mains pour n’yrentrer peut-etre jamais, nous ajouterons par la suite dans les supplemens de l’Encyclopediece qui aura ete ajoute a la theorie de la lune, depuis le mois de Novembre 1759, ou nousecrivons cet article.”Pour calculer l’avance de l’apogee, d’Alembert commence par utiliser la formule approcheede u(z) que nous avons donnee plus haut. En annulant la derivee de u par rapport a zet en utilisant le fait qu’en une apogee zap, le nombre reel Nzap est proche d’un multipleentier de 2π, il trouve l’equation approchee

sin(Nzap) ∼19

sin(2zap − 2nzap).

Il obtient ainsi une avance moyenne de l’apogee de 1o30′ modulee par une “equation”periodique d’une amplitude de 6o20′ qui est l’angle dont le sinus vaut 1/9. Il constate alorsque les Principia donnent le double pour la valeur observee de l’avance moyenne mais aussipour l’equation, avec de plus dans ce cas une difference de signe. La discussion de ce qu’ilen est vraiment de l’“equation” est delicate et je renvoie aux notes 89 et 106 de MadameChapront-Touze pour une discussion complete. Disons seulement que d’Alembert calculel’apogee de l’orbite de la Lune (en projection sur l’ecliptique) alors que Newton etudie lesmouvements “seculaires” d’une ellipse approchee. Les 6o20′ de d’Alembert correspondenten gros a l’addition de deux “inegalites” :– d’une part l’equation semestre de Newton dont l’amplitude (observee et non calculeepar Newton) est de 12o18′ (ramene aujourd’hui a 9o39′) et le signe oppose a celui ded’Alembert ; c’est une inegalite de type “seculaire”, faisant intervenir un angle a longueperiode, l’angle du Soleil avec l’apogee de la Lune ;

7

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1963 г. ноябрь—декабрь т. XVIII, вып. 6 (114) УСПЕХИ МАТЕМАТИЧЕСКИХ НАУК

МАЛЫЕ ЗНАМЕНАТЕЛИ И ПРОБЛЕМЫ УСТОЙЧИВОСТИ ДВИЖЕНИЯ В КЛАССИЧЕСКОЙ И НЕБЕСНОЙ МЕХАНИКЕ

В. П. А р н о л ь д

СОДЕРЖАНИЕ

В в е д е н и е 92 § 1. Результаты 92 § 2. Предварительные сведения из механики 93 § 3. Предварительные сведения из математики 94 § 4. Простейшая проблема устойчивости 97 § 5 . Содержание работы 100

Г л а ва I. Теория возмущений 100 § 1. Интегрируемые и неинтегрируемые проблемы динамики 101 § 2. Классическая теория возмущений 102 § 3. Малые знаменатели 103 § 4. Метод Ньютона , 104 § 5. Собственное вырождение 106 § 6. Замечание 1 108 § 7. Замечание 2 110 § 8. Применение к задаче о собственном вырождении 112 § 9. Предельное вырождение. Преобразование Биркгофа . . . . . . . . 113 § 10. Устойчивость положений равновесия гамильтоновых систем , . . . 115

Г л а в а II. Адиабатические инварианты 117 § 1. Понятие адиабатического инварианта 117 § 2. Вечная адиабатическая инвариантность действия при медл ином пе­

риодическом изменении функции Гамильтона 119 § 3. Адиабатические инварианты консервативных систем 123 § 4. Магнитные ловушки 126 § 5. Многомерный случай 129

Г л а в а III. Об устойчивости планетных движений 130 § 1. Картина движения 130 § 2. Переменные Якоби, Делоне и Пуанкаре 134 § 3. Преобразование Биркгофа 136 § 4. Вычисление асимптотики коэффициентов разложения Fi 138 § 5. Задача многих тел 143

Г л а в а IV. Основная теорема 146 § 1. Основная теорема 147 § 2. Индуктивная теорема 148 § 3 . Индуктивная лемма . 149

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242 J. Laskar Seminaire Poincare

nouvelle machine, nous avons pu utiliser 5 Mh en 2008, ce qui a permis de finaliserles calculs.

2.6 Possibilite de collisions entre Mercure, Mars, Venus et la Terre

Fig. 6 – Exemple d’evolution a long terme des orbites des planetes telluriques : Mercure (blanc),Venus (vert), Terre (bleu), Mars (rouge). Le temps est indique en milliers d’annees (kyr). (a) Auvoisinage de l’etat actuel, les orbites se deforment sous l’influence des perturbations planetaires,mais sans permettre de rencontres proches ou de collisions. (b) Dans pres de 1% des cas, l’orbitede Mercure peut se deformer suffisamment pour permettre une collision avec Venus ou le Soleil enmoins de 5 Ga. (c) Pour l’une des trajectoires, l’excentricite de Mars augmente suffisamment pourpermettre une rencontre proche ou une collision avec la Terre. (d) Ceci entraıne une destabilisationdes planetes telluriques qui permet aussi une collision entre Venus et la Terre (Figure issue desresultats des simulations numeriques de Laskar et Gastineau, 2009).

Grace a la machine JADE, nous avons pu simuler 2501 solutions differentes dumouvement des planetes du Systeme solaire sur 5 milliards d’annees, correspondanta l’esperance de vie du systeme, avant que le Soleil ne devienne une geante rouge.

Les 2501 solutions calculees sont compatibles avec notre connaissance actuelledu Systeme solaire. Dans la majorite des cas, celui-ci continue d’evoluer commedans les quelques millions d’annees actuels : les orbites planetaires se deforment etprecessent sous l’influence des perturbations mutuelles des planetes mais sans pos-sibilite de collisions ou d’ejections de planetes hors du Systeme solaire. Neanmoins,dans 1% des cas environ, l’excentricite de Mercure augmente considerablement. Dansplusieurs cas, cette deformation de l’orbite de Mercure conduit alors a une collision

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©2002 Donald Kahn

School of Mathematics Richard McGehee

Seminar

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-6 -4 -2 0 2 4 6-6-4

-2 0

2 4

6-3 -2 -1 0 1 2 3-6

-4-2

0 2

4 6

L4

L5

J

Tuesday, June 21, 2011

Appendice 2. La note aux C.R.A.S. de Poincare (30 novembre 1896).

18

Figure 7: The Schubart solution for equal masses(animation by Rick Moeckel). Remember that the

conservation of energy implies infinite velocities at collision

Figure 8: The figure eight (Animation by Rick Moeckel).

In the planetary problem, Poincaré used various techniques, in particular continuation, to prove the existence of various families of

periodic (or periodic modulo rotations) orbits. He defined sorts, genres, species. In the first sort, the eccentricities of the planets are small

and they have no inclination; in the limit where the masses vanish, the orbits become circular, with rationally dependent frequencies. In

the second sort, the inclinations are still zero but the eccentricities are finite; in the limit one gets elliptic motions with the same direction

of major semi-axes and conjunctions or oppositions at each half-period. In the third sort, eccentricities are small but inclinations are

finite and the limit motions are circular but inclined. Solutions of the second genre are called today subharmonics: they are associated to

a given -periodic solution of one of the 3 sorts and their period is an integer multiple of . Solutions of the second species are

particularly interesting: in the limit of zero masses, the planets follow Keplerian orbits till they have a close encounter (i.e., in the limit, a

collision) and then shift to another pair of Keplerian ellipses. A full symbolic dynamics of such almost collision orbits has been

constructed by Bolotin: it implies the existence of solutions with an erratic diffusion of the angular momentum and a much slower one of

the Jacobi constant.

Numerical exploration

In the twentieth century, extensive search for families of periodic solutions

in the restricted 3-body problem was accomplished, first by mechanical

quadratures at the Copenhagen Observatory (Stromgren), later using

computers by Hénon at the Nice Observatory, Broucke, and others. The

books by Hénon [He] and Bruno [Br] describe both theoretically and

numerically the so-called generating families of periodic solutions of the

planar circular restricted 3-body problem (i.e. the limits of the families

when the mass of one of the massive bodies tends to zero). An extensive

study of the phase space of the related Hill's problem was completed by

Simo and Stuchi. Particularly interesting for mission design are the Halo

orbits in the spatial restricted problem, which bifurcate from a planar

Lyapunov family originating from a collinear relative equilibrium. Other

much studied special cases are the collinear problem with the remarkable periodic (regularized) solution discovered by Schubart and the

isosceles problem, where one body moves on a line, while the two others, with the same mass, move symmetrically on the orthogonal

line (resp. plane in the spatial case)

Stability, exponents, invariant manifolds

Poincaré initiated also the study of the stability of periodic orbits, introducing their characteristic exponents and their stable and

unstable manifolds. The famous mistake in his 1889 prize memoir, where he had thought he had proved stability in the restricted

problem, is about the intersection of these manifolds (see [BG]). Let us recall that the collinear relative equilibria are always linearly

unstable; for the equilateral ones, linear stability occurs only when one mass greatly dominates the two others (Routh[5] (http://www-

groups.dcs.st-and.ac.uk/~history/Biographies/Routh.html) criterion, see Trieste notes (http://www.math.umn.edu/~rick/notes/Notes.html)

) as in the case of Sun-Jupiter-Trojans already alluded to. The study of the intersections of stable and unstable manifolds of known

periodic solutions leads to the construction of orbits by methods of symbolic dynamics.

Minimizing the action

Solutions of the equations of motion are critical

paths of the Lagrangian action ,

defined on the Sobolev space of paths with

value in the configuration space of the problem. Among these,

the minimizers are likely to be the simplest and Poincaré

proposed to look for them in a short note of 1896. Coercivity

(i.e. forbidding minimizers at infinity) is achieved by restricting

appropriately. Thanks to Tonelli (http://it.wikipedia.org

/wiki/Leonida_Tonelli) 's theorem this insures the existence of a

minimizer with values in the closure of , that is of a path

possibly with collisions. Indeed, Newton's force is weak enough

so that the action remains finite along a path which ends in a

configuration where some of the bodies are in collision.

Technically, this comes from Sundman estimates

for any pair of bodies entering a collision at time . The equilateral

homographic solutions (of any eccentricity) are characterized as action minimizers in their homotopy class among loops of fixed period

Three body problem - Scholarpedia http://www.scholarpedia.org/article/Three_body_problem

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Three body problem

From Scholarpedia

Alain Chenciner (2007), Scholarpedia, 2(10):2111. doi:10.4249/scholarpedia.2111 revision #79311 [link to/cite this article]

Hosting and maintenance of this article is sponsored by Brain Corporation.

Curator: Dr. Alain Chenciner, Math Dept Paris 7 University and IMCCE (Paris Observatory), France

The problem is to determine the possible motions of three point masses , , and , which attract each other according to

Newton's (http://www-history.mcs.st-andrews.ac.uk/Biographies/Newton.html) law of inverse squares. It started with the perturbative

studies of Newton himself on the inequalities of the lunar motion[1] (http://en.wikisource.org

/wiki/Philosophiae_Naturalis_Principia_Mathematica/Preface) . In the 1740s it was constituted as the search for solutions (or at least

approximate solutions) of a system of ordinary differential equations by the works of Euler (http://www-history.mcs.st-andrews.ac.uk

/Biographies/Euler.html) , Clairaut (http://www-history.mcs.st-andrews.ac.uk/Biographies/Clairaut.html) and d'Alembert (http://www-

history.mcs.st-andrews.ac.uk/Biographies/D'Alembert.html) (with in particular the explanation by Clairaut of the motion of the lunar

apogee). Much developed by Lagrange (http://www-history.mcs.st-andrews.ac.uk/Biographies/Lagrange.html) , Laplace (http://www-

history.mcs.st-andrews.ac.uk/Biographies/Laplace.html) and their followers, the mathematical theory entered a new era at the end of the

19th century with the works of Poincaré (http://www-history.mcs.st-andrews.ac.uk/Biographies/Poincare.html) and since the 1950's with

the development of computers. While the two-body problem is integrable and its solutions completely understood (see [2]

(http://en.wikipedia.org/wiki/Two-body_problem) ,[AKN],[Al],[BP]), solutions of the three-body problem may be of an arbitrary

complexity and are very far from being completely understood.

Contents

1 Equations

2 Symmetries, first integrals

3 Homographic solutions

4 The astronomer's three-body problem: i) the planetary problem

4.1 Reduction to the general problem of dynamics

4.2 The secular system

4.3 From Lindstedt series to K.A.M.

5 The astronomer's three-body problem: ii) a caricature of the lunar problem

5.1 The planar circular restricted problem:

5.2 The simplest case:

5.3 Poincaré's first return map

5.4 Higher values of the Jacobi constant

6 Periodic solutions

6.1 Poincaré's classification

6.2 Numerical exploration

6.3 Stability, exponents, invariant manifolds

6.4 Minimizing the action

7 Global evolution

7.1 Lagrange-Jacobi and Sundman

7.2 The shape sphere

7.3 Collisions

7.4 Final motions

7.5 The oldest open question in dynamical systems

8 Non-integrability

8.1 Bruns, Painlevé

8.2 Poincaré

8.3 Ziglin, Morales-Ramis

8.4 Two cases of integrability

9 Still simpler than the 4- (and more)-body problem !

Three body problem - Scholarpedia http://www.scholarpedia.org/article/Three_body_problem

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