An *inertia matrix* (also called an *inertia tensor* or *inertia operator*) will be derived by considering as a collection
of particles that are rigidly attached together (all contact forces
between them cancel due to Newton's third law).
The expression
in (13.77) represents the
mass of an infinitesimal particle of . The *moment of
momentum* of the infinitesimal particle is
. This means that the total moment of momentum of is

By using the fact that , the expression becomes

Observe that now appears twice in the integrand. By doing some algebraic manipulations, can be removed from the integrand, and a function that is quadratic in the variables is obtained (since is a vector, the function is technically a quadratic form). The first step is to apply the identity to obtain

The angular velocity can be moved to the right to obtain

in which the integral now occurs over a matrix and is the identity matrix.

Let be called the *inertia matrix* and be defined as

Using the definition,

This simplification enables a concise expression of (13.82) as

which makes use of the chain rule.

Steven M LaValle 2020-08-14