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. Dirichlet 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 Dirichlet …"

Excerpt from p. 530 of R. Baker, C. Christenson & H. Orde (2004) which, in turn, is taken from R. Dedekind's accounting of The life of Bernhard Riemann:

"In the Easter vacation of 1860 [Riemann] went on a trip to Paris, where he stayed for a month from 26th March; unfortunately the weather was raw and unfriendly and in the last week of his visit there was one day after another of snow and hail which made it almost impossible to see the sights. However, he was delighted with the friendly reception which he received from the Parisian scholars Serret, Bertrand, Hermite, Puiseux and Briot, with whom he spent a pleasant day in the country at Chatenay, along with Bouquet. In the same year, [Riemann] completed his paper on the motion of a fluid ellipsoid …"

Excerpt from pp. 184-185 of EFE:

"Riemann's paper 'Ein Beitrag zu den Untersuchungen über die Bewegung eines flüssigen gleichartigen Ellipsoides,' communicated to Der Königlichen Gesellschaft der Wissenschaften zu Göttingen on December 8, 1860, is remarkable for the wealth of new results it contains and for the breadth of its comprehension of the entire range of problems. In the present writer's [S. Chandrasekhar] view this much neglected paper — [for example] there are no references to it in any of the writings of Poincaré, Darwin, or Jeans — deserves to be included among the other great papers of Riemann that are well known. … In view of Riemann's unique place in science, a critical appraisal of this paper is perhaps justified."

His Published Work on 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).

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

Derivation

Step 1

At the beginning of his discussion, Riemann denotes by $(x,y,z)$ "the [inertial-frame] coordinates of an element of the fluid body at time $t$," and he denotes by $(\xi ,\eta ,\zeta )$ "the coordinates of the point $(x,y,z)$ with respect to a moving coordinate system, whose axes coincide at each instant with the principal axes of the ellipsoid." Drawing from our accompanying discussion of Euler angles, it seems appropriate to associate ${\overrightarrow{A}}_{XYZ}$ with the vector that points from the origin to the $(x,y,z)$ location of the fluid element as viewed from the inertial reference frame, and to associate ${\overrightarrow{A}}_{\mathrm{b}\mathrm{o}\mathrm{d}\mathrm{y}}$ with the vector that points from the origin to the same location of the fluid element, but as viewed from Riemann's specified rotating frame. With this in mind, we have the following notation mappings:

${\overrightarrow{A}}_{\mathrm{X}\mathrm{Y}\mathrm{Z}}(A_X,A_Y,A_Z)$	$\to$	${\overrightarrow{A}}_{\mathrm{i}\mathrm{n}\mathrm{e}\mathrm{r}\mathrm{t}\mathrm{i}\mathrm{a}\mathrm{l}}(x,y,z)={\overrightarrow{e}}_X(x)+{\overrightarrow{e}}_Y(y)+{\overrightarrow{e}}_Z(z),$
${\overrightarrow{A}}_{\mathrm{b}\mathrm{o}\mathrm{d}\mathrm{y}}(A_1,A_2,A_3)$	$\to$	${\overrightarrow{A}}_{\mathrm{b}\mathrm{o}\mathrm{d}\mathrm{y}}(\xi ,\eta ,\zeta )={\overrightarrow{e}}_1(\xi )+{\overrightarrow{e}}_2(\eta )+{\overrightarrow{e}}_3(\zeta ).$

Again drawing from our accompanying Euler angles discussion, quite generally these two coordinate representations — of the same vector, $\overrightarrow{A}$ — are related to one another via the matrix expression,

${\overrightarrow{A}}_{\mathrm{b}\mathrm{o}\mathrm{d}\mathrm{y}}=\hat{R}\cdot {\overrightarrow{A}}_{\mathrm{i}\mathrm{n}\mathrm{e}\mathrm{r}\mathrm{t}\mathrm{i}\mathrm{a}\mathrm{l}},$

that is,

$\left[\begin{array}{c}\xi \\ \eta \\ \zeta \end{array}\right]$

$=$

$\left[\begin{array}{ccc}{\overrightarrow{e}}_1\cdot {\overrightarrow{e}}_X& {\overrightarrow{e}}_1\cdot {\overrightarrow{e}}_Y& {\overrightarrow{e}}_1\cdot {\overrightarrow{e}}_Z\\ {\overrightarrow{e}}_2\cdot {\overrightarrow{e}}_X& {\overrightarrow{e}}_2\cdot {\overrightarrow{e}}_Y& {\overrightarrow{e}}_2\cdot {\overrightarrow{e}}_Z\\ {\overrightarrow{e}}_3\cdot {\overrightarrow{e}}_X& {\overrightarrow{e}}_3\cdot {\overrightarrow{e}}_Y& {\overrightarrow{e}}_3\cdot {\overrightarrow{e}}_Z\end{array}\right]\cdot \left[\begin{array}{c}x\\ y\\ z\end{array}\right]=\hat{R}\cdot \left[\begin{array}{c}x\\ y\\ z\end{array}\right].$

In terms of the trio of Euler angles, the rotation matrix is,

$\hat{R}(\varphi ,\theta ,\psi )$

$=$

$\left[\begin{array}{ccc}(\cos \psi \cos \varphi -\sin \varphi \sin \psi \cos \theta )& (\cos \psi \sin \varphi +\cos \varphi \sin \psi \cos \theta )& (\sin \psi \sin \theta )\\ (-\sin \psi \cos \varphi -\sin \varphi \cos \psi \cos \theta )& (-\sin \varphi \sin \psi +\cos \varphi \cos \psi \cos \theta )& (\cos \psi \sin \theta )\\ (\sin \theta \sin \varphi )& (-\sin \theta \cos \varphi )& (\cos \theta )\end{array}\right].$

This means that — see our accompanying discussion — in terms of the trio of Euler angles, the following three coordinate mappings hold:

$\xi$	=	$x(\cos \psi \cos \varphi -\sin \psi \sin \varphi \cos \theta )+y(\sin \varphi \cos \psi +\sin \psi \cos \theta \cos \varphi )+z(\sin \psi \sin \theta ),$
$\eta$	=	$x(-\sin \psi \cos \varphi -\sin \varphi \cos \theta \cos \psi )+y(-\sin \psi \sin \varphi +\cos \psi \cos \theta \cos \varphi )+z(\sin \theta \cos \psi ),$
$\zeta$	=	$x(\sin \theta \sin \varphi )+y(-\sin \theta \cos \varphi )+z(\cos \theta ).$

Notice the correspondence between this set of coordinate relations and the set marked by Riemann (1861) as equation (2) of §1.

From §1 of Riemann (1861)

Then $\xi$, $\eta$, $\zeta$ are known … to be linear expressions in $x$, $y$, $z$, $\xi$ = $\alpha x+\beta y+\gamma z,$ $\eta$ = ${\alpha }^{\prime }x+{\beta }^{\prime }y+{\gamma }^{\prime }z,$ $\zeta$ = ${\alpha }^{″}x+{\beta }^{″}y+{\gamma }^{″}z.$ The coefficients are the cosines of the angles that the axes of one system form with the axes of the other …

Alternatively, given that,

${\overrightarrow{A}}_{\mathrm{i}\mathrm{n}\mathrm{e}\mathrm{r}\mathrm{t}\mathrm{i}\mathrm{a}\mathrm{l}}={\hat{R}}^{-1}\cdot {\overrightarrow{A}}_{\mathrm{b}\mathrm{o}\mathrm{d}\mathrm{y}},$

the following additional three mapping relations also must hold:

$x$	=	$\xi (\cos \psi \cos \varphi -\sin \psi \sin \varphi \cos \theta )+\eta (-\sin \psi \cos \varphi -\sin \varphi \cos \theta \cos \psi )+\zeta (\sin \theta \sin \varphi ),$
$y$	=	$\xi (\sin \varphi \cos \psi +\sin \psi \cos \theta \cos \varphi )+\eta (-\sin \psi \sin \varphi +\cos \psi \cos \theta \cos \varphi )+\zeta (-\sin \theta \cos \varphi ),$
$z$	=	$\xi (\sin \psi \sin \theta )+\eta (\sin \theta \cos \psi )+\zeta (\cos \theta ).$

Step 2

Given that $(x,y,z)$ and $(\xi ,\eta ,\zeta )$ are components of the same position vector, $\overrightarrow{A}$, it must in general be the case that the square of the length of the vector is the same for both coordinate representations. That is, as Riemann (1861) states in §1 following equation (2), it must be the case that,

${\xi }^2+{\eta }^2+{\zeta }^2=x^2+y^2+z^2.$

Let's specifically assess whether or not this holds true when Euler angles are used to relate the components of the two position vector expressions.

${\xi }^2+{\eta }^2+{\zeta }^2$	=	$[x(\cos \psi \cos \varphi -\sin \psi \sin \varphi \cos \theta )+y(\sin \varphi \cos \psi +\sin \psi \cos \theta \cos \varphi )+z(\sin \psi \sin \theta ){]}^2$
		$+[x(-\sin \psi \cos \varphi -\sin \varphi \cos \theta \cos \psi )+y(-\sin \psi \sin \varphi +\cos \psi \cos \theta \cos \varphi )+z(\sin \theta \cos \psi ){]}^2$
		$+[x(\sin \theta \sin \varphi )+y(-\sin \theta \cos \varphi )+z(\cos \theta ){]}^2.$

On the right-hand side, the coefficient of various terms is …

$x^2:$	$(\cos \psi \cos \varphi -\sin \psi \sin \varphi \cos \theta {)}^2+(-\sin \psi \cos \varphi -\sin \varphi \cos \theta \cos \psi {)}^2+(\sin \theta \sin \varphi {)}^2$
	$={\cos }^2\psi {\cos }^2\varphi -2\cos \psi \cos \varphi \sin \psi \sin \varphi \cos \theta +{\sin }^{}\psi {\sin }^{}\varphi {\cos }^{}\theta +{\sin }^{}\psi {\cos }^{}\varphi +2\sin \psi \cos \varphi \sin \varphi \cos \theta \cos \psi +{\sin }^{}\varphi {\cos }^{}\theta {\cos }^{}\psi +{\sin }^{}\theta {\sin }^{}\varphi$
	$=(2\sin \psi \cos \varphi \sin \varphi \cos \theta \cos \psi -2\cos \psi \cos \varphi \sin \psi \sin \varphi \cos \theta )+({\sin }^{}\psi +{\cos }^{}\psi ){\sin }^{}\varphi {\cos }^{}\theta +({\sin }^{}+{\cos }^{}\psi ){\cos }^{}\varphi +{\sin }^{}\theta {\sin }^{}\varphi$
	$={\sin }^2\varphi {\cos }^2\theta +{\cos }^2\varphi +{\sin }^2\theta {\sin }^2\varphi$
	$=1.$

$y^2:$	$(\sin \varphi \cos \psi +\sin \psi \cos \theta \cos \varphi {)}^2+(-\sin \psi \sin \varphi +\cos \psi \cos \theta \cos \varphi {)}^2+(-\sin \theta \cos \varphi {)}^2$
	$={\sin }^2\varphi {\cos }^2\psi +2\sin \varphi \cos \psi \sin \psi \cos \theta \cos \varphi +{\sin }^{}\psi {\cos }^{}\theta {\cos }^{}\varphi +{\sin }^{}\psi {\sin }^{}\varphi -2\sin \psi \sin \varphi \cos \psi \cos \theta \cos \varphi +{\cos }^{}\psi {\cos }^{}\theta {\cos }^{}\varphi +{\sin }^{}\theta {\cos }^{}\varphi$
	$=(2\sin \varphi \cos \psi \sin \psi \cos \theta \cos \varphi -2\sin \psi \sin \varphi \cos \psi \cos \theta \cos \varphi )+({\cos }^{}\psi +{\sin }^{}\psi ){\sin }^{}\varphi +({\sin }^{}\psi +{\cos }^{}\psi ){\cos }^{}\theta {\cos }^{}\varphi +{\sin }^{}\theta {\cos }^{}\varphi$
	$={\sin }^2\varphi +{\cos }^2\theta {\cos }^2\varphi +{\sin }^2\theta {\cos }^2\varphi$
	$=1.$

$z^2:$	$(\sin \psi \sin \theta {)}^2+(\sin \theta \cos \psi {)}^2+(\cos \theta {)}^2$
	$=1.$

$2xy:$	$(\cos \psi \cos \varphi -\sin \psi \sin \varphi \cos \theta )(\sin \varphi \cos \psi +\sin \psi \cos \theta \cos \varphi )+(-\sin \psi \cos \varphi -\sin \varphi \cos \theta \cos \psi )(-\sin \psi \sin \varphi +\cos \psi \cos \theta \cos \varphi )+(\sin \theta \sin \varphi )(-\sin \theta \cos \varphi )$
	$=\cos \psi \cos \varphi (\sin \varphi \cos \psi +\sin \psi \cos \theta \cos \varphi )-\sin \psi \sin \varphi \cos \theta (\sin \varphi \cos \psi +\sin \psi \cos \theta \cos \varphi )+(\sin \psi \cos \varphi +\sin \varphi \cos \theta \cos \psi )(\sin \psi \sin \varphi -\cos \psi \cos \theta \cos \varphi )-{\sin }^{}\theta \sin \varphi \cos \varphi$
	$=\cos \psi \cos \varphi (\sin \varphi \cos \psi +\sin \psi \cos \theta \cos \varphi )-\sin \psi \sin \varphi \cos \theta (\sin \varphi \cos \psi +\sin \psi \cos \theta \cos \varphi )+(\sin \psi \cos \varphi +\sin \varphi \cos \theta \cos \psi )(\sin \psi \sin \varphi -\cos \psi \cos \theta \cos \varphi )-{\sin }^{}\theta \sin \varphi \cos \varphi$

See Also

• C. G. J. Jacobi (1834, Annalen der Physik, vol. 109, Issue 8, pp. 229-233), Ueber die Figur des Gleichgewichts
• 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)