The tensor equation \(AX +XA = \Phi (A,H)\), with applications to kinematics of continua (Q1805067)

A linear equation \(AX+ XA= \Phi(A, H)\), where \(A\), \(X\) and \(H\) are linear transformations of a two- or three-dimensional inner product space \(V\), and \(\Phi(A, H)\) an isotropic function of \(A\) and \(H\) which is linear in \(H\), is considered for certain \(\Phi\). Qualitative properties of the solution \(X\) and relations between the solutions for various forms of \(\Phi\) are established (this is done for an inner product space of arbitrary dimension). These results are applied in case \(\dim V= 3\) to obtain new explicit formulas for \(X\) as well as to derive some previously known formulas. Several applications to the kinematics of continua are considered.