On vector constants of motion for dynamical systems admittingsymmetry groups of the inverse cube law equation

摘要

It is shown in H. White, Nuovo Cimento B, 121 (2006) 111, that anydynamical system possessing the Lie symmetry group of the equation x = -Gr-4xas a symmetry group, is of the form x = Kir-4x + K2r-3L + K3r-4x L, whereL = x A X, and the K, are functions of L =ILI. We show that such systems admita vector constant of motion of the form A = r 1H1x + H2L + r-1H3x n L, wherethe Hz are also functions of L, which satisfy the first-order system K3Hf = LH3,K3(LH2)' = -K2H3, K3(LH3)' = H2K2 - H1. In the case K3 0 we show that, if(r, 6, 0) are spherical coordinates with A parallel to the line 0 = 0, then there existsa function u(L) of L such that z = u(L)+0 is a constant of motion of the system. Weshow further that given any solution (H1, H2, H3) to the above linear system, two(linear independent) solutions (Hf, Hf, H3 )p = 1, 2 can be constructed such thatthe corresponding vector constants Al and A2 are mutually orthogonal, orthogonalto A and satisfy the relations Al = cos zul + sin zu2, A2 = - sin zul + cos zu2,where the u' are fixed orthogonal unit vectors in the plane perpendicular to A. Weobserve that in the case K3 = 0, L is constant, H1 = 112 K2 and that A is a constantmultiple of the Poincare vector L+K2r-1x. We note that with the exception of therotation symmetry any Lie point symmetry (of the equation x = -Gr-4x) is alsoa symmetry of A and show that A admits symmetries other than Lie symmetries.