A convenient expression of the time-derivative zn(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

DOI

10.1080/14029251.2016.1237197How to use a DOI?

Keywords

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

Abstract

Let p_N (z; t) be a (monic) time-dependent polynomial of arbitrary degree N in z, and let z_n ≡ z_n (t) be its N zeros: pN (z;t)=∏n=1N[z-zn(t)] . In this paper we report a convenient expression of the k-th time-derivative zn(k)(t) of the zero z_n (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”).

Copyright

© 2016 The Authors. Published by Atlantis Press and Taylor & Francis

Open Access

This is an open access article distributed under the CC BY-NC 4.0 license (http://creativecommons.org/licenses/by-nc/4.0/).

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TY  - JOUR
AU  - Mario Bruschi
AU  - Francesco Calogero
PY  - 2021
DA  - 2021/01/06
TI  - A convenient expression of the time-derivative 

zn(k)(t)

, of arbitrary order k, of the zero zn(t) of a time-dependent polynomial pN(z;t) of arbitrary degree N in z, and solvable dynamical systems
JO  - Journal of Nonlinear Mathematical Physics
SP  - 474
EP  - 485
VL  - 23
IS  - 4
SN  - 1776-0852
UR  - https://doi.org/10.1080/14029251.2016.1237197
DO  - 10.1080/14029251.2016.1237197
ID  - Bruschi2021
ER  -