A convenient expression of the time-derivative z(n)((k))(t), of arbitrary order k, of the zero z(n)(t) of a time-dependent polynomial p(N)(z;t) of arbitrary degree N in z, and solvable dynamical systems

作者：Bruschi Mario^*; Calogero Francesco

来源：Journal of Nonlinear Mathematical Physics, 2016, 23(4): 474-485.

DOI：10.1080/14029251.2016.1237197

• time-derivatives of time-dependent polynomials
• time-derivatives of the zeros of time-dependent polynomials
• solvable dynamical systems
• solvable many-body problems
• isochronous many-body problems

Let p(N) (z;t) be a (monic) time-dependent polynomial of arbitrary degree N in z; and let z(n) equivalent to z(n) (t) be its N zeros: p(N)(z;t) = Pi(N)(n-1)[z - z(n) (t)]. In this paper we report a convenient expression of the k-th time-derivative z(n)((k))(t) of the zero zn (t). This formula plays a key role in the identification of classes of solvable dynamical systems describing the motion of point-particles moving in the complex z-plane while nonlinearly interacting among themselves; one such example, featuring many arbitrary parameters, is reported, including its variation describing the motion of many particles moving in the real Cartesian xy-plane and interacting among themselves via rotation-invariant Newtonian equations of motion ("accelerations equal forces").

• 出版日期2016