Reduced Dynamics from the Unitary Group to Some Flag Manifolds :   Interacting Matrix Riccati Equations
  In this paper we treat the time evolution of unitary elements in the N level system and consider the reduced dynamics from the unitary group U(N) to flag manifolds of the second type (in our terminology). Then we derive a set of differential equations of matrix Riccati types interacting with one another and present an important problem on a nonlinear superposition formula that the Riccati equation satisfies.   Our result is a natural generalization of the paper {\bf Chaturvedi et al} (arXiv : 0706.0964 [quant-ph]). 
Kazuyuki Fujii (Yokohama City University) and Hiroshi Oike