LettreGeorge Howard Darwin à Henri Poincaré - 20 mars 1899

3-15-3. George Howard Darwin to H. Poincaré

20.3.99

Newnham Grange–Cambridge

Dear Monsieur Poincaré,

I think you will be interested to learn that I have at last discovered (as I believe with confidence) the natural history, so to speak, of the A orbits & the figure-of-8 which I will call A.11Following up on a previous letter to Poincaré of 28.02.1899 (§ 3-15-1), Darwin returns here to his classification of periodic orbits (1897), which Poincaré considered in the third volume of the Méthodes nouvelles de la mécanique céleste (Poincaré 1899, 352). Somewhere between C=40 & 39.6 an orbit of ejection which is periodic arises of this form:

As C falls in value the loop diminishes becomes a cusp & then becomes rounded & coalesces with A when both vanish.

For a very slightly smaller value of C another orbit of ejection arises of this form:

In this case also the loop diminishes, becomes a cusp, then rounded & we pass on to the figures-of-8 A which I have traced.

I missed these families because they have so transitory an existence. I have my idea of how they will behave as to stability.

Now pass to the side of J remote from S & it is obvious that the C orbit ends in an orbit of ejection thus:

For a very slightly smaller value of C a new orbit of ejection arises thus:

(I should have drawn the other half of these.)

This orbit is obviously the parent of a new series of figures-of-8 thus

The quasi-coalescence & disappearance of the figure-of-eight A (or say A) with B may take place through the figure-of-8 terminating in an orbit of ejection, & B with another. But I have not yet the materials for deciding this point. These ideas throw a flood of light on the march of the orbits, but I will not trouble you with more.

I think I shall ask Mittag Leffler to let me have another paper in the Acta but I fear it will be some months before I shall be ready.22Darwin published his thoughts only a decade later (Darwin 1909). A few years after Darwin sent this letter to Poincaré, Hough (1901) found a family of figure-of-eight orbits that Darwin had missed (Barrow-Green 1997, 196).

I remain, Yours sincerely

G. H. Darwin

ALS 4p. Collection particulière, Paris 75017.

References

Titre (dcterms:title)

George Howard Darwin à Henri Poincaré - 20 mars 1899

Incipit (ahpo:incipit)

I think you will be interested to learn ...

Date (ahpo:writingDate)

1899-03-20

Expéditeur (ahpo:sentBy)

Destinataire (ahpo:sentTo)

Adresse (ahpo:destinationAddress)

Lieu (ahpo:writtenAt)

Chapitre (ahpo:publishedIn)

Lieu d’archivage (ahpo:archivedAt)

Type (ahpo:documentType)

fr Lettre autographe signée

Section (dans le livre) (ahpo:sectionNumber)

3

Identifiant dans les archives locales (ahpo:identifierInLocalArchives)

CD n° 113

Droits (ahpo:rightsHolder)

Archives Henri Poincaré

Nombre de pages (ahpo:numberOfPages)

4

Langue (ahpo:language)

en

Publié sous la référence (ahpo:publishedInReference)

Gharnati Ph.D.

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« George Howard Darwin à Henri Poincaré - 20 Mars 1899 ». La Correspondance Entre Henri Poincaré, Les Astronomes Et Les géodésiens. Archives Henri Poincaré, s. d, Archives Henri Poincaré, s. d, La correspondance d'Henri Poincaré, consulté le 30 janvier 2023, http://henripoincare.fr/s/correspondance/item/10291