Twirl tensors and the tensor equation AX−XA=C

Abstract

We call t) the twirl tensor. Hoger [2] used the term "twirl" for fir, which rotates the Lagrangian triad of orthonormal eigenvectors of the right stretch tensor U. In a similar manner we can define twirl tensors for other mechanical and physical quantities (such as the stretch tensors U, V, the Cauchy stress tensor tt, the inertia tensor of a rigid body, etc.) which can be characterized by a second-order symmetric tensor. For definiteness in our discussion, we shall take the twirl tensor f~ as the representative of all possible twirl tensors. An expression in "principal axis" notation was given in [ 1] for t). In this paper we seek an intrinsic expression for tl. To this end we first derive a tensor equation that fl should satisfy. Indeed, if we can justifiably differentiate the spectral decomposition