Journal of Nonlinear Mathematical Physics

Volume 23, Issue 4, September 2016, Pages 474 - 485

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

Authors
Mario Bruschi, Francesco Calogero
Physics Department, University of Rome “La Sapienza”, Italy
Istituto Nazionale di Fisica Nucleare, Sezione di Roma, Italy
Received 22 February 2016, Accepted 3 July 2016, Available Online 6 January 2021.
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 pN (z; t) be a (monic) time-dependent polynomial of arbitrary degree N in z, and let znzn (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 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”).

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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Journal
Journal of Nonlinear Mathematical Physics
Volume-Issue
23 - 4
Pages
474 - 485
Publication Date
2021/01/06
ISSN (Online)
1776-0852
ISSN (Print)
1402-9251
DOI
10.1080/14029251.2016.1237197How to use a DOI?
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/).

Cite this article

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  -