Central force and the higher rank numerical range

Let r=r(θ)r=r(θ) be the orbit of a point mass under a central force f(r)=−1/r3f(r)=−1/r3 with angular momentum M  . Suppose p=M/M2−1. We show that the orbit is a transcendental curve if p   is irrational, and the orbit is an algebraic curve FA(1,x,y)=0FA(1,x,y)=0 for some 2m×2m2m×2m nilpotent Toeplitz matrix A   if p=m/j0p=m/j0 is rational, whereF(t,x,y)=FA(t,x,y)=det(tIn+x(A+A⁎)/2+y(A−A⁎)/(2i)).F(t,x,y)=FA(t,x,y)=det(tIn+x(A+A⁎)/2+y(A−A⁎)/(2i)). Furthermore, we examine the rank-k   numerical range Λk(A)Λk(A) of this nilpotent Toeplitz matrix, showing that the sum of numbers of flat portions on the boundary of Λk(A)Λk(A), k=1,2,…,mk=1,2,…,m, is (m−1)(2m−3)(m−1)(2m−3).