3Dconfigurations/RiemannEllipsoids

Riemann (1826 - 1866)

Background

Excerpt from p. 539 of R. Baker, C. Christenson & H. Orde (2004):

"Newton first showed that the departure of the figure of the earth from a sphere is due to its rotation. Jacobi showed in 1834 that gravitational equilibrium of a rotating spheroid is consistent with three distinct axes if the angular momentum exceeds a critical value. Direchlet had posed and partially analyzed the conditions for a configuration which is an ellipsoid varying with time, such that the motion in an inertial frame, is linear in the coordinates. His results were edited posthumously by Dedekind in 1860. In his published work dated 1861 — five years before his death — Riemann took up this problem of Direchlet …"

Riemann Ellipsoids

Riemann's (1861) work, titled, "Ein Beitrag zu den Untersuchungen über die Bewegung eines flüssigen gleichartigen Ellipsoides" — English translation: "A contribution to the study of the motion of a homogeneous fluid ellipsoid" — can be found in various published collections of his papers:

• In the German language:   Bernhard Riemann (1876) Gesammelte Mathematische Werke und Wissenschaftlicher, especially Chapter X (p. 168).
• In the German language:   "Bernhard Riemann's Gesammelte Mathematische Werke," 2^nd edition, edited by Heinrich Weber, Teubner, Leipzig, 1892.
• In English:   "Bernhard Riemann Collected Papers," translated by Roger Baker, Charles Christenson and Henry Orde from the 1892 (German) edition and published in 2004 (Heber City, Utah, USA: Kendrick Press).

The description and detailed analysis of Riemann's (1861) work that follows, below, draws primarily from the 2004 translation of his collected works.

See Also

• Jacobi (1834)
• Dedekind (1860)
• Bernhard Riemann (1876) Gesammelte Mathematische Werke und Wissenschaftlicher, especially Chapter X (p. 168) titled (something along the following line), "A Contribution to Research on Rotating Ellipsoidal Fluids"
• S. Chandrasekhar (1965), ApJ, 142, 890 - 961. The Equilibrum and the Stability of the Riemann Ellipsoids. I. — This work is referenced as Paper XXV in EFE and focuses on S-type Riemann ellipsoids.
• S. Chandrasekhar (1966), ApJ, 145, 842 - 877. The Equilibrum and the Stability of the Riemann Ellipsoids. II. — This work is referenced as Paper XXVIII in EFE and focuses on Riemann ellipsoids of Types I, II and III.
• Chandrasekhar & Lebovitz (1990)