Rigid Body Rotations and Euler's Equations
Physics 212x
Harrison Hartle

In the original form of classical mechanics developed by Isaac Newton, the dynamics of objects was modeled primarily by making a large simplification: treating objects as point particles with no spatial extent. This is a viable assumption for modeling many systems such as the dynamics of planetary bodies. However, when attempting to understand the rotational and translational motion of objects of size comparable to the distances of their movements, the point particle model fails. In cases where the relative configuration of an object’s makeup is fixed, that object is called a rigid body and the mechanics governing it’s motion are called rigid body dynamics.

A rigid body has a fixed internal configuration of particles, continuously distributed. A rigid body has six degrees of freedom: three dimensions of translation and three angles of rotation. The complete orientation of a rigid body can be specified by those six numbers. A number of issues need to be addressed in order to model the motion of extended objects, in describing the angular orientation of the body, choosing a convenient reference frame, and modeling the object’s moment of inertia in three dimensions. Some of the basic techniques for addressing these issues are discussed here.

Fixed Frames and Body Frames

Moment of Inertia, Principal Moments of Inertia, Inertia Tensor

Relationships Between Vectors in Inertial and Rotating Frames

Derivation of Euler's Equations

Bibliography