A remark on dynamical systems with Poincare vectors

摘要

We show that the dynamical system x = pix + p_2L + ps(x A L), x G Rs, wheie L = x A x and the pi are functions of x and x, admits a vector constant of motion A satisfying the Poincare vector condition A x = Cr, where C is a nonzero constant if and only if p2 = GLT~3 where L = L, and G is some nonzero constant in which case A = C(G-1L + r-1x) where L = L~lL, (When C = 0, the condition becomes P2 = 0 and the vector constant of motion is any constant multiple of L.) We note that for such a system L = -p^r2L, i.e.. L need not be constant for dynamical systems admitting such a vector constant of motion. (In the case L is constant, G can also be any function of L.) Some details witlrlf information on the equations of the orbit curves of such systems are given.