On the N -extended Euler system: generalized Jacobi elliptic functions

We study the integrable system of first order differential equations ω _i(v)'=α _i ∏ _j iω _j(v) , (1≤ i, j≤ N) as an initial value problem, with real coefficients α _i and initial conditions ω _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 dv^*=g(ω _i) dv that separate geometry from dynamic. Some of those parametrizations turn out to be generalization of the Jacobi amplitude .