Geometry of Rotation: Understanding the Dynamics of Rigid Bodies

To fulfil the need for a suitable mathematical language, Geometry of Rotation presents: chapter 1 on vector algebra, chapter 2 on the algebra of dyadics and chapter 3 on transformations, invariance and the calculus of vectors and dyadics. Geometry of Rotation presents a new comprehensive theory on rigid body angular displacement. Chapter 4 introduces the concepts of the rotation vector and the rotation dyadic for simple rotations (about a fixed direction). Chapter 5 begins with a set of three experiments on finite successive rotations of a rigid body and ends with a presentation of the Euler angles as three properly defined finite successive rotations. Chapter 6 presents rigid body angular displacement theory in vector-dyadic language. The rotation vector emerges as the vector embedded in the rotation dyadic with several remarkable and very useful properties. Formulas are obtained for the resultant rotation vector of two and of three successive finite rotations.

Geometry of Rotation incorporates this angular displacement theory and presents an up-to-date, comprehensive theory of the kinematics of rigid bodies. Chapter 7 presents the kinematic problem of the rotation of a rigid body about a fixed point and proof that the angular velocity vector is the time derivative of the rotation vector. In chapter 8, the concepts of the rotation vector, the rotation dyadic and the angular velocity vector are generalized to cases of rigid bodies in relative rotational motion. Chapter 8 includes the derivation of formulas for the composition of the kinematic state vectors for multi-body systems of rigid bodies coupled at points to each other. Finally, chapter 9, presents the case of the general motion of the unconstrained rigid body, cases of generally coupled rigid bodies in general motion, and a complete set of extendable self-consistent kinematic state vector equations for generally coupled multi-body systems.