A convenient expression of the time-derivative, 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. Issue 4 (1st October 2016)

• Record Type: Journal Article
• Authors: Bruschi, Mario; Calogero, Francesco
• Is Part Of: Journal of nonlinear mathematical physics. Volume 23:Issue 4(2016)
• Page Start: 474
• Page End: 485
• Publication Date: 2016-10-01
• Subjects: 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; Nonlinear theories -- Periodicals; Mathematical physics -- Periodicals; 515.252
• DOI: 10.1080/14029251.2016.1237197
• Languages: English
• ISSNs: 1402-9251

Abstract

Abstract : Let p N ( z ; t ) be a (monic) time-dependent polynomial of arbitrary degree N in z, and let z n ≡ zn ( t ) be its N zeros: . In this paper we report a convenient expression of the k -th time-derivative 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").