Time-dependent polynomials with one multiple root and new solvable dynamical systems

TL;DR

This paper introduces new solvable dynamical systems derived from time-dependent polynomials with a multiple root, expanding the class of algebraically solvable N-body problems with explicit examples of isochronous systems.

Contribution

It develops a novel method to relate derivatives of roots to polynomial coefficients, leading to new solvable first-order ODE systems and N-body problems with multiple roots.

Findings

Constructed new algebraically solvable first-order ODE systems.

Provided explicit examples of isochronous 2- and 3-body problems.

Extended the class of solvable dynamical systems with multiple roots.

Abstract

A time-dependent monic polynomial in the z variable with N distinct roots such that exactly one root has multiplicity m>=2 is considered. For k=1,2, the k-th derivatives of the N roots are expressed in terms of the derivatives of order j<= k of the first N coefficients of the polynomial and of the derivatives of order j<= k-1 of the roots themselves. These relations are utilized to construct new classes of algebraically solvable first order systems of ODEs as well as N-body problems. Multiple examples of solvable isochronous (all solutions are periodic with the same period) 2- and 3-body problems are provided.