On the $$\varvec{N}$$ N -extended Euler system: generalized Jacobi elliptic functions

Abstract

We study the integrable system of first order differential equations \(\omega _i(v)'=\alpha _i\,\prod _{j\ne i}\omega _j(v)\), \((1\le i, j\le N)\) as an initial value problem, with real coefficients \(\alpha _i\) and initial conditions \(\omega _i(0)\). The geometrical structure of the system allows to express it as a Poisson system. The analysis is based on its quadratic first integrals. For each dimension N, the system defines a family of functions, generically hyperelliptic functions. When \(N=3\), this system generalizes the classic Euler system for the reduced flow of the free rigid body problem; thus, we call it N-extended Euler system (N-EES). In this paper, the cases \(N=4\) and \(N=5\) are studied, generalizing Jacobi elliptic functions which are defined as a 3-EES. The \(N=4\) case was proposed in Hille (Lectures on Ordinary Differential Equations. Addison-Wesley, Reading, 1969), and the solution is presented in Abdel-Salam (Z Naturforsch A 64a:639–645, 2009; it is still expressed as elliptic functions. The hyperelliptic functions arise for the \(N=5\) case, which also contain special solutions in elliptic form. Taking into account the nested structure of the N-EES, we propose reparametrizations of the type \({\mathrm{d}}v^*=g(\omega _i)\,{\mathrm{d}}v\) that separate geometry from dynamic. Some of those parametrizations turn out to be generalization of the Jacobi amplitude.