In astrophysics, Dirichlet's ellipsoidal problem, named after Peter Gustav Lejeune Dirichlet, asks under what conditions there can exist an ellipsoidal configuration at all times of a homogeneous rotating fluid mass in which the motion, in an inertial frame, is a linear function of the coordinates.
Dirichlet's basic idea was to reduce Euler equations to a system of ordinary differential equations such that the position of a fluid particle in a homogeneous ellipsoid at any time is a linear and homogeneous function of initial position of the fluid particle, using Lagrangian framework instead of the Eulerian framework.
[1][2][3] In the winter of 1856–57, Dirichlet found some solutions of Euler equations and he presented those in his lectures on partial differential equations in July 1857 and published the results in the same month.
[4] His work was left unfinished at his sudden death in 1859, but his notes were collated and published by Richard Dedekind posthumously in 1860.
[5] Bernhard Riemann said, "In his posthumous paper, edited for publication by Dedekind, Dirichlet has opened up, in a most remarkable way, an entirely new avenue for investigations on the motion of a self-gravitating homogeneous ellipsoid.
The further development of his beautiful discovery has a particular interest to the mathematician even apart from its relevance to the forms of heavenly bodies which initially instigated these investigations."
Dirichlet's problem is generalized by Bernhard Riemann in 1860[6] and by Norman R. Lebovitz in modern form in 1965.
be the semi-axes of the ellipsoid, which varies with time.
Since the ellipsoid is homogeneous, the constancy of mass requires the constancy of the volume of the ellipsoid, same as the initial volume.
We can define an anti-symmetric matrix with this, where we can write the dual
By definition, Dirichlet's problem is looking for a solution which is a linear function of initial condition
Let us assume the following form, and we define a diagonal matrix
with diagonal elements being the semi-axes of the ellipsoid, then above equation can be written in matrix form as where
transforms one unit vector on the boundary to another unit vector on the boundary, in other words, it is orthogonal, i.e.,
The Dirichlet's ellipsoidal problem then reduces to finding whether the matrix
and that it is expressible in terms of two orthogonal matrices as in
be the velocity field seen by the observer at rest in the moving frame, which can be regarded as the internal fluid motion since this excludes the uniform rotation seen by the inertial observer.
This internal motion is found to given by whose components, explicitly, are given by These three components show that the internal motion is composed of two parts: one with a uniform vorticity
with components and the other with a stagnation point flow, i.e.,
The pressure is found to assume a quadratic form, as derived by the equation of motion (and using the vanishing condition at the surface) given by where
Substituting this back in the equation of motion leads to where
is diagonal matrix, whose diagonal elements are given by The tensor momentum equation and the conservation of mass equation, i.e.,
is admissible under the conditions of Dirichlet's problem, then the motion determined by the transpose
In other words, the theorem can be stated as for any state of motions that preserves a ellipsoidal figure, there is an adjoint state of motions that preserves the same ellipsoidal figure.
By taking transpose of the tensor momentum equation, one sees that the role of
The following relations confirms the previous statement.
where, further The typical configuration of this theorem is the Jacobi ellipsoid and its adjoint is called as Dedekind ellipsoid, in other words, both ellipsoid have same shape, but their internal fluid motions are different.
The tensor momentum equation admits three integrals, with regards to conservation of energy, angular momentum and circulation.
, from the above integral, we obtain where we substituted the formula for
, where the circulation components (in the inertial frame) are given by