16.5 :  角運動量と慣性主軸

The angular momentum for a rigid body can be expressed as the integral of the cross-product of the position vector of the mass element with the cross-product of the angular velocity of the body and the position vector.

This equation can be written in terms of rectangular coordinates by choosing another set of xyz axes that are inclined arbitrarily with the reference frame.

The rectangular components of angular momentum are derived by expanding the cross-product, combining i, j, and k terms, and applying the product of inertia definition.

These equations can be simplified further by choosing the xyz axes such that they form principal axes for the rigid body.

In this case, the rectangular components of the angular momentum are expressed in terms of the principal moments of inertia about xyz axes.

Each of these components of angular momentum is independent of the other and follows the principle of conservation of angular momentum independently.