On the extended Euler system and the Jacobi and Weierstrass elliptic functions

Francisco  Crespo; Sebastián  Ferrer

doi:10.3934/jgm.2015.7.151

Article Contents

2015, Volume 7, Issue 2: 151-168. Doi: 10.3934/jgm.2015.7.151

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On the extended Euler system and the Jacobi and Weierstrass elliptic functions

Francisco Crespo^1, and
Sebastián Ferrer^1,

1.
Departamento de Matemática Aplicada, Universidad de Murcia, 30071 Espinardo

Received: March 31, 2014

Revised: January 31, 2015

Published: May 2015

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Abstract

We study the extended Euler systems (EES) as an initial value problem. Particular realizations of it lead to several Lie-Poisson structures. We consider a 6-D Poisson structure that fit all of them together. The symplectic stratification of this non Lie-Poisson structure uses the first integrals which are elliptic and hyperbolic cylinders, although other quadrics may be used as well. A qualitative study of the solutions is carried out and the twelve Jacobi elliptic functions in the real domain are shown in an unified way as the solutions of the EES. As a consequence, Jacobi's transformation for the elliptic modulus is obtained. Likewise, introducing the square norm function we establish in a straightforward way the connection of the EES with the Weierstrass $\wp$ elliptic function, giving the relation of its invariants $g_i$ with the integrals and coefficients of the EES.

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Mathematics Subject Classification: Primary: 33E05, 34A34; Secondary: 37K10, 70H06.

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