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 AA orbits & the figure-of-8 which I will call AA^{\prime}.11endnote: 1 Following 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=40C=40 & 39.639.6 an orbit of ejection which is periodic arises of this form:

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

For a very slightly smaller value of CC 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 AA^{\prime} 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 JJ remote from SS & it is obvious that the CC orbit ends in an orbit of ejection thus:

For a very slightly smaller value of CC 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 AA (or say AA^{\prime}) with BB may take place through the figure-of-8 terminating in an orbit of ejection, & BB 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.22endnote: 2 Darwin 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.

Time-stamp: " 4.05.2019 00:12"

Notes

  • 1 Following 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).
  • 2 Darwin 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).

References

  • J. E. Barrow-Green (1997) Poincaré and the Three Body Problem. AMS/LMS, Providence. Cited by: endnote 2.
  • G. H. Darwin (1897) Periodic orbits. Acta mathematica 21, pp. 101–242. Link Cited by: endnote 1.
  • G. H. Darwin (1909) On certain families of periodic orbits. Monthly Notices of the Royal Astronomical Society 70, pp. 108–143. Link Cited by: endnote 2.
  • S. S. Hough (1901) On certain discontinuities connected with periodic orbits. Acta mathematica 24, pp. 257–288. Link Cited by: endnote 2.
  • H. Poincaré (1899) Les méthodes nouvelles de la mécanique céleste, Volume 3. Gauthier-Villars, Paris. Link Cited by: endnote 1.