Overview

In this chapter we return to transformations among Cartesian frames of reference. In the first chapter we studied physics in inertial and linearly accelerated frames. Here, we formulate physics in rotating frames from both the Newtonian and Lagrangian standpoints.

Once one separates the translational motion of the center of mass then rigid body theory for a single solid body is the theory of rigid rotations in three dimensions. The different coordinate systems needed for describing rigid body motions are the different possible parameterizations of the rotation group. The same tools are used to describe transformations among rotating and nonrotating frames in particle mechanics, and so we begin with the necessary mathematics. We introduce rotating frames and the motions of bodies relative to those frames in this chapter and go on to rigid body motions in the next. Along the way, we will need to diagonalize matrices and so we include that topic here as well.

In discussing physics in rotating frames, we are prepared to proceed in one of two ways: by a direct term by term transformation of Newton's laws to rotating frames, or by a direct application of Lagrange's equations, which are covariant. We follow both paths in this chapter because it is best to understand physics from as many different angles as possible. Before using Lagrange's equations, we also derive a formulation of Newton's equations that is covariant with respect to transformations to and among rotating Cartesian frames.