newtonian mechanics - Can any physical rigid body be represented by an ellipsoid with the same angular dynamics?

According to wikipedia, the inertia tensor of an ellipsoid with semi-axes $a,b,c$ and mass $m$ is

$$\left[\begin{array}{ccc} \frac{m}{5}(b^2+c^2)&0&0\\ 0&\frac{m}{5}(a^2+c^2)&0\\ 0&0&\frac{m}{5}(a^2+b^2)\\ \end{array}\right].$$

If you create an arbitrary 3x3 positive diagonal matrix and try to solve for the $a,b,c$, it's very easy to wind up with imaginary dimensions. If I try to place separate point masses, I seem to run into the same problem.

Does that mean that the tensor doesn't represent a physically possible distribution of mass, or just not a uniform density solid? Intuitively, at least, it seems that it must be impossible for an inertia tensor to a have a single large value and two small values since a single point mass with a non-zero radius will always affect two dimensions equally and an ring of infinitesimal height still leaves the two minor dimensions with half the momentum of the large principal axis.

Answer

1. 
   

   Proposition: Given an arbitrary rigid body (and wrt. to an arbitrary choice of pivotal point for the rigid body, and wrt. to an arbitrary choice of Cartesian coordinates $x$, $y$, and $z$), then the diagonal elements $I_{xx}$, $I_{yy}$, and $I_{zz}$ of the inertia tensor satisfy the triangle inequality, $$ I_{xx} +I_{yy} ~\geq~ I_{zz}, \qquad I_{yy} +I_{zz} ~\geq~ I_{xx}, \qquad I_{zz} +I_{xx} ~\geq~ I_{yy}. \tag{1} $$ Sketched proof: Write down the definition of moment of inertia.$\Box$

   
   


2. 
   

   Observation: It follows from the triangle inequality (1) alone that $$I_{xx}, I_{yy},I_{zz}~\geq~0 \tag{2}$$ are non-negative. (The ineq. (2) of course also follows from the definition of moment of inertia.)

   
   


3. 
   

   Corollary of Proposition: Given an arbitrary rigid body (and wrt. to an arbitrary choice of pivotal point for the rigid body), then the three moments of inertia $I_x$, $I_y$, and $I_z$, around the three principal axes (which we will call $x$, $y$, and $z$) satisfy the triangle inequality, $$ I_x +I_y ~\geq~ I_z, \qquad I_y +I_z ~\geq~ I_x, \qquad I_z +I_x ~\geq~ I_y. \tag{3} $$

   
   



4. 
   

   In other words, if a semi-positive definite symmetric real $3\times 3$ matrix with non-negative eigenvalues $I_x$, $I_y$, and $I_z$ does not satisfy the triangle inequality (3), it doesn't represent a physically possible distribution of mass.

   
   


5. 
   

   Conversely, one may show that given three eigenvalues $I_x$, $I_y$, and $I_z$ that satisfy (3), they may be reproduced by a solid ellipsoid with a unique choice of non-negative semi-axes $a$, $b$, and $c$ (unique up to the scaling of the total mass $m$). $$ \frac{2}{5}m a^2~=~I_y +I_z -I_x~\geq~0, $$ $$ \frac{2}{5}m b^2~=~I_z +I_x -I_y~\geq~0, $$ $$ \frac{2}{5}m c^2~=~I_x +I_y -I_z~\geq~0.\tag{4} $$