A rigid body is an example of a mechanical system with constraints. Thus, in a sense this chapter on rigid bodies was nothing but an extended example of the application of the ideas developed in the first chapter. The equations of motion are just the Lagrange equations.
The kinetic energy for a rigid body separates into a translational kinetic energy and a rotational kinetic energy. The center of mass plays a special role in this separation. The rotational kinetic energy is simply expressed in terms of the inertia tensor and the angular velocity vector. We developed the expressions for the kinetic energy that take into account the body constraints, and we expressed the remaining degrees of freedom in terms of suitable generalized coordinates.
The vector angular momentum is conserved if there are no external torques. The time derivative of the body components of the angular momentum can be written entirely in terms of the body components of the angular momentum, and the three principal moments of inertia. The body components of angular momentum form a self-contained dynamical subsystem.
One choice for generalized coordinates is the Euler angles. They form suitable generalized coordinates, but are otherwise not special or well motivated. The Lagrange equations for the Euler angles are singular for some Euler angles. Other choices of generalized coordinates like the Euler angles have similar problems. Equations of motion for the orientation vector are nonsingular.
In general the potential energy depends on the details of the mass distribution, and does not separate as the kinetic energy separated into center of mass and relative contributions.
For an axisymmetric top with uniform gravitational acceleration, the potential energy is exactly the potential energy due to elevation of the center of mass. Aspects of the motion of the top are deduced from the conserved quantities. Euler angles are just the right thing for this problem.
For other problems, such as the rotational motion of an out-of-round satellite near a planet, the potential energy cannot be written in finite terms, and judicious approximations must be made. The essential character of such diverse systems as the rotation of the Moon, Hyperion, and Mercury are captured by a simple model problem.