# American Institute of Mathematical Sciences

June  2015, 7(2): 151-168. doi: 10.3934/jgm.2015.7.151

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

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

Received  April 2014 Revised  February 2015 Published  June 2015

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.
Citation: Francisco Crespo, Sebastián Ferrer. On the extended Euler system and the Jacobi and Weierstrass elliptic functions. Journal of Geometric Mechanics, 2015, 7 (2) : 151-168. doi: 10.3934/jgm.2015.7.151
##### References:

show all references

##### References:
 [1] Sebastián Ferrer, Francisco J. Molero. Andoyer's variables and phases in the free rigid body. Journal of Geometric Mechanics, 2014, 6 (1) : 25-37. doi: 10.3934/jgm.2014.6.25 [2] Kai Koike. Wall effect on the motion of a rigid body immersed in a free molecular flow. Kinetic & Related Models, 2018, 11 (3) : 441-467. doi: 10.3934/krm.2018020 [3] Bernard Bonnard, Olivier Cots, Nataliya Shcherbakova. The Serret-Andoyer Riemannian metric and Euler-Poinsot rigid body motion. Mathematical Control & Related Fields, 2013, 3 (3) : 287-302. doi: 10.3934/mcrf.2013.3.287 [4] Jingyu Li. Asymptotic behavior of solutions to elliptic equations in a coated body. Communications on Pure & Applied Analysis, 2009, 8 (4) : 1251-1267. doi: 10.3934/cpaa.2009.8.1251 [5] Sujit Nair, Naomi Ehrich Leonard. Stable synchronization of rigid body networks. Networks & Heterogeneous Media, 2007, 2 (4) : 597-626. doi: 10.3934/nhm.2007.2.597 [6] Peter Bella, Arianna Giunti. Green's function for elliptic systems: Moment bounds. Networks & Heterogeneous Media, 2018, 13 (1) : 155-176. doi: 10.3934/nhm.2018007 [7] Lucas C. F. Ferreira, Everaldo Medeiros, Marcelo Montenegro. An elliptic system and the critical hyperbola. Communications on Pure & Applied Analysis, 2015, 14 (3) : 1169-1182. doi: 10.3934/cpaa.2015.14.1169 [8] Fabiana Maria Ferreira, Francisco Odair de Paiva. On a resonant and superlinear elliptic system. Discrete & Continuous Dynamical Systems - A, 2019, 39 (10) : 5775-5784. doi: 10.3934/dcds.2019253 [9] Shingo Takeuchi. The basis property of generalized Jacobian elliptic functions. Communications on Pure & Applied Analysis, 2014, 13 (6) : 2675-2692. doi: 10.3934/cpaa.2014.13.2675 [10] Hugo Beirão da Veiga. Elliptic boundary value problems in spaces of continuous functions. Discrete & Continuous Dynamical Systems - S, 2016, 9 (1) : 43-52. doi: 10.3934/dcdss.2016.9.43 [11] Marcel Guardia, Tere M. Seara, Pau Martín, Lara Sabbagh. Oscillatory orbits in the restricted elliptic planar three body problem. Discrete & Continuous Dynamical Systems - A, 2017, 37 (1) : 229-256. doi: 10.3934/dcds.2017009 [12] Samuel R. Kaplan, Ernesto A. Lacomba, Jaume Llibre. Symbolic dynamics of the elliptic rectilinear restricted 3--body problem. Discrete & Continuous Dynamical Systems - S, 2008, 1 (4) : 541-555. doi: 10.3934/dcdss.2008.1.541 [13] Giuseppe Maria Coclite, Helge Holden, Kenneth H. Karlsen. Wellposedness for a parabolic-elliptic system. Discrete & Continuous Dynamical Systems - A, 2005, 13 (3) : 659-682. doi: 10.3934/dcds.2005.13.659 [14] Rafael Ortega, James R. Ward Jr. A semilinear elliptic system with vanishing nonlinearities. Conference Publications, 2003, 2003 (Special) : 688-693. doi: 10.3934/proc.2003.2003.688 [15] Daniela De Silva, Fausto Ferrari, Sandro Salsa. On two phase free boundary problems governed by elliptic equations with distributed sources. Discrete & Continuous Dynamical Systems - S, 2014, 7 (4) : 673-693. doi: 10.3934/dcdss.2014.7.673 [16] Hua Chen, Wenbin Lv, Shaohua Wu. A free boundary problem for a class of parabolic-elliptic type chemotaxis model. Communications on Pure & Applied Analysis, 2018, 17 (6) : 2577-2592. doi: 10.3934/cpaa.2018122 [17] Daniela De Silva, Fausto Ferrari, Sandro Salsa. Recent progresses on elliptic two-phase free boundary problems. Discrete & Continuous Dynamical Systems - A, 2019, 0 (0) : 1-18. doi: 10.3934/dcds.2019239 [18] E. N. Dancer, Danielle Hilhorst, Shusen Yan. Peak solutions for the Dirichlet problem of an elliptic system. Discrete & Continuous Dynamical Systems - A, 2009, 24 (3) : 731-761. doi: 10.3934/dcds.2009.24.731 [19] Dongho Chae. Existence of a semilinear elliptic system with exponential nonlinearities. Discrete & Continuous Dynamical Systems - A, 2007, 18 (4) : 709-718. doi: 10.3934/dcds.2007.18.709 [20] Diane Denny. A unique positive solution to a system of semilinear elliptic equations. Conference Publications, 2013, 2013 (special) : 193-195. doi: 10.3934/proc.2013.2013.193

2018 Impact Factor: 0.525

## Metrics

• HTML views (0)
• Cited by (4)

• on AIMS