Gradient flows and double bracket equations

A unified extension of the gradient flows and the double bracket equations of Chu–Driessel and Brockett is obtained in the frame work of reductive Lie groups. We examine the gradient flows on the orbit in the Cartan subspace of a reductive Lie algebra, under the adjoint action. The results of Chu–Driessel and Brockett are corresponding to the reductive groups GL(n,R) and O(p,q).