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) - Bruschi, Mario; Calogero, Francesco
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").