Example 13
Matrices Applied to 
the Moment of Inertia Tensor

Copyright © 1993 by Bill Titus, Carleton College, 
Department of Physics and Astronomy, Northfield, MN  55057-4025
September 6, 1993

Topics and Skills

Problem

Solution - Part 1

Step 1 - Generate a symbolic 3 x 3 inertia tensor  iTensor  using the MMA command  Array[].  Denote the symbolic elements of  iTensor  by  iT.

Step 2 -  Evaluate the nine elements of the inertia tensor for this particular problem.  Do the necessary 2D integrations symbolically using  Integrate[].  Display the results in matrix form.  You should find that  iTensor  equals

Comment 2

Solution - Part b

Step 1 - One can determine the eigenvalues  e  of the inertia tensor by finding the roots of the characteristic polynomial,  
det(iTensor - eI), where  I  is the identify matrix.  First find the characteristic polynomial as a function of  e  and call it  charPoly.  The identity matrix in MMA is generated by the function  IdentityMatrix[].  How do you think one calculates the determinant of a matrix within MMA?

Step 2 - Your polynomial probably looks a bit ugly.  Try factoring it using MMA's function  Factor[].

Step 3 - Find the three roots of your polynomial equation using  Solve[].  Call the result  eigenValues.

Step 4 - The  eigenvector  vj  corrresponding to the eigenvalue  ej  satisfies the matrix equation

                       (iTensor  -  ej  I) vj  =  0

and corresonds to one of the principle axes of rotation.  Because these equations are homogeneous, there will be one arbitrary component in each eigenvector; this component is sometimes fixed by requiring that the eigenvector be normalized.  The following code determines the unnormalized components of the eigenvector corresponding to the first eigenvalue.  Look it over and then activate the cell.  Finally modify the code to find the other two eigenvectors.

Clear[vVector, v, myEqu]
vVector = {v[1], v[2], v[3]};
myEqu = (iTensor - eigenValues[[1,1,2]] *
         IdentityMatrix[3]) . vVector;
Solve[myEqu[[1]] == 0, v[1]]
Solve[myEqu[[2]] == 0, v[1]]
Solve[myEqu[[3]] == 0, v[3]]

Comment 4

Step 5 - There's an easier way to find the eigenvalues and eigenvectors, and that's by letting MMA to it for you.  Find the MMA functions which determine eigenvalues and eigenvectors and apply them to your problem.

Comment 5

Step 5 -  Since the inertia tensor is real and symmetric, its eigenvectors should be orthogonal.  Use MMA to prove that your eigenvectors are indeed orthogonal.

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