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

Previous Article
On the relation between geometrical quantum mechanics and information geometry
JGM Home
This Issue
Next Article
Completeness properties of Sobolev metrics on the space of curves

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

Francisco Crespo ^1, and Sebastián Ferrer ^1,

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

Received April 2014 Revised February 2015 Published June 2015

Abstract
Related Papers

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.

Keywords: Weierstrass elliptic function., Euler system, Jacobi elliptic functions, Free rigid body with torques.

Mathematics Subject Classification: Primary: 33E05, 34A34; Secondary: 37K10, 70H0.

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:

[1]	P. F. Byrd and M. D. Friedman, Handbook of Elliptic Integrals for Engineers and Scientists, 2nd edition, Springer-Verlag, Berlin Heidelberg New York, 1971.
[2]	F. Crespo, Hopf Fibration Reduction of a Quartic Model. An Application to Rotational and Orbital Dynamics, Ph.D thesis, Universidad de Murcia, 2015, 208 pp.
[3]	S. Ferrer, F. Crespo and F. J. Molero, On the N-extended Euler system I. Generalized Jacobi elliptic functions, submitted to Nonlinear Dynamics, arXiv:1505.06142, 2015.
[4]	S. Ferrer and F. Crespo, On a quartic polynomial model. Parametric quartic Hamiltonian model. A unified treatment of classic integrable systems, Journal of Geometric Mechanics, 6 (2014), 479-502. doi: 10.3934/jgm.2014.6.479.
[5]	A. G. Greenhill, The Applications of Elliptic Functions, Macmillan and Co., London, 1892.
[6]	C. Gudermann, Theorie der Modular Functionen,, Crelle's Journal, ().
[7]	J. K. Hale, Ordinary Differential Equations, Wiley-Interscience, New York, 1969.
[8]	D. D. Holm and J. E. Marsden, The rotor and the pendulum, in Symplectic Geometry and Mathematical Physics, Prog. In Math., 99, Birkhäuser Boston, Boston, MA, 1991, 189-203.
[9]	T. Iwai and D. Tarama, Classical and quantum dynamics for an extended free rigid body, Differential Geometry and its Applications, 28 (2010), 501-517. doi: 10.1016/j.difgeo.2010.05.002.
[10]	D. F. Lawden, Elliptic Functions and Aplications, Vol. 80, Springer-Verlag, New York, 1989. doi: 10.1007/978-1-4757-3980-0.
[11]	J. E. Marsden and T. S. Ratiu, Introduction to Mechanics and Symmetry, 2nd edition, Springer-Verlag, New York, 1999. doi: 10.1007/978-0-387-21792-5.
[12]	K. R. Meyer, Jacobi elliptic functions from a dynamical system point of view, The American Mathematical Monthly, 108 (2001), 729-737. doi: 10.2307/2695616.
[13]	F. J. Molero, M. Lara, S. Ferrer and F. Céspedes, 2-D Duffing oscillator: Elliptic functions from a dynamical system point of view, Qual. Theory Dyn. Syst., 12 (2013), 115-139. doi: 10.1007/s12346-012-0081-1.
[14]	Y. Nambu, Generalized Hamiltonian mechanics, Phys. Rev., 7 (1973), 2405-2412. doi: 10.1103/PhysRevD.7.2405.
[15]	M. Puta, On the dynamics of the rigid body with two torques, C. R. Acad. Sci. Paris, 317 (1993), 377-380.
[16]	M. Puta, Stability and control in spacecraft dynamics, Journal of Lie Theory, 7 (1997), 269-278.
[17]	M. Puta and I. Casu, Geometrical aspects in the rigid body dynamics with three quadratic controls, in Geometry, Integrability and Quantization (Varna, 1999), Coral Press Sci. Publ., Sofia, 2000, 209-224.
[18]	A. Weinstein, The local structure of Poisson manifolds, J. Differential Geom., 18 (1983), 523-557.
[19]	E. T. Whittaker and G. N. Watson, A Course of Modern Analysis, Cambridge Mathematical Library, Cambridge University Press, Cambridge, 1996. doi: 10.1017/CBO9780511608759.

show all references

References:

[1]	P. F. Byrd and M. D. Friedman, Handbook of Elliptic Integrals for Engineers and Scientists, 2nd edition, Springer-Verlag, Berlin Heidelberg New York, 1971.
[2]	F. Crespo, Hopf Fibration Reduction of a Quartic Model. An Application to Rotational and Orbital Dynamics, Ph.D thesis, Universidad de Murcia, 2015, 208 pp.
[3]	S. Ferrer, F. Crespo and F. J. Molero, On the N-extended Euler system I. Generalized Jacobi elliptic functions, submitted to Nonlinear Dynamics, arXiv:1505.06142, 2015.
[4]	S. Ferrer and F. Crespo, On a quartic polynomial model. Parametric quartic Hamiltonian model. A unified treatment of classic integrable systems, Journal of Geometric Mechanics, 6 (2014), 479-502. doi: 10.3934/jgm.2014.6.479.
[5]	A. G. Greenhill, The Applications of Elliptic Functions, Macmillan and Co., London, 1892.
[6]	C. Gudermann, Theorie der Modular Functionen,, Crelle's Journal, ().
[7]	J. K. Hale, Ordinary Differential Equations, Wiley-Interscience, New York, 1969.
[8]	D. D. Holm and J. E. Marsden, The rotor and the pendulum, in Symplectic Geometry and Mathematical Physics, Prog. In Math., 99, Birkhäuser Boston, Boston, MA, 1991, 189-203.
[9]	T. Iwai and D. Tarama, Classical and quantum dynamics for an extended free rigid body, Differential Geometry and its Applications, 28 (2010), 501-517. doi: 10.1016/j.difgeo.2010.05.002.
[10]	D. F. Lawden, Elliptic Functions and Aplications, Vol. 80, Springer-Verlag, New York, 1989. doi: 10.1007/978-1-4757-3980-0.
[11]	J. E. Marsden and T. S. Ratiu, Introduction to Mechanics and Symmetry, 2nd edition, Springer-Verlag, New York, 1999. doi: 10.1007/978-0-387-21792-5.
[12]	K. R. Meyer, Jacobi elliptic functions from a dynamical system point of view, The American Mathematical Monthly, 108 (2001), 729-737. doi: 10.2307/2695616.
[13]	F. J. Molero, M. Lara, S. Ferrer and F. Céspedes, 2-D Duffing oscillator: Elliptic functions from a dynamical system point of view, Qual. Theory Dyn. Syst., 12 (2013), 115-139. doi: 10.1007/s12346-012-0081-1.
[14]	Y. Nambu, Generalized Hamiltonian mechanics, Phys. Rev., 7 (1973), 2405-2412. doi: 10.1103/PhysRevD.7.2405.
[15]	M. Puta, On the dynamics of the rigid body with two torques, C. R. Acad. Sci. Paris, 317 (1993), 377-380.
[16]	M. Puta, Stability and control in spacecraft dynamics, Journal of Lie Theory, 7 (1997), 269-278.
[17]	M. Puta and I. Casu, Geometrical aspects in the rigid body dynamics with three quadratic controls, in Geometry, Integrability and Quantization (Varna, 1999), Coral Press Sci. Publ., Sofia, 2000, 209-224.
[18]	A. Weinstein, The local structure of Poisson manifolds, J. Differential Geom., 18 (1983), 523-557.
[19]	E. T. Whittaker and G. N. Watson, A Course of Modern Analysis, Cambridge Mathematical Library, Cambridge University Press, Cambridge, 1996. doi: 10.1017/CBO9780511608759.

[1]	Banavara N. Shashikanth. Poisson brackets for the dynamically coupled system of a free boundary and a neutrally buoyant rigid body in a body-fixed frame. Journal of Geometric Mechanics, 2020, 12 (1) : 25-52. doi: 10.3934/jgm.2020003
[2]	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
[3]	Kai Koike. Wall effect on the motion of a rigid body immersed in a free molecular flow. Kinetic and Related Models, 2018, 11 (3) : 441-467. doi: 10.3934/krm.2018020
[4]	Bernard Bonnard, Olivier Cots, Nataliya Shcherbakova. The Serret-Andoyer Riemannian metric and Euler-Poinsot rigid body motion. Mathematical Control and Related Fields, 2013, 3 (3) : 287-302. doi: 10.3934/mcrf.2013.3.287
[5]	Jingyu Li. Asymptotic behavior of solutions to elliptic equations in a coated body. Communications on Pure and Applied Analysis, 2009, 8 (4) : 1251-1267. doi: 10.3934/cpaa.2009.8.1251
[6]	Sujit Nair, Naomi Ehrich Leonard. Stable synchronization of rigid body networks. Networks and Heterogeneous Media, 2007, 2 (4) : 597-626. doi: 10.3934/nhm.2007.2.597
[7]	Peter Bella, Arianna Giunti. Green's function for elliptic systems: Moment bounds. Networks and Heterogeneous Media, 2018, 13 (1) : 155-176. doi: 10.3934/nhm.2018007
[8]	Fabiana Maria Ferreira, Francisco Odair de Paiva. On a resonant and superlinear elliptic system. Discrete and Continuous Dynamical Systems, 2019, 39 (10) : 5775-5784. doi: 10.3934/dcds.2019253
[9]	Lucas C. F. Ferreira, Everaldo Medeiros, Marcelo Montenegro. An elliptic system and the critical hyperbola. Communications on Pure and Applied Analysis, 2015, 14 (3) : 1169-1182. doi: 10.3934/cpaa.2015.14.1169
[10]	Marcel Guardia, Tere M. Seara, Pau Martín, Lara Sabbagh. Oscillatory orbits in the restricted elliptic planar three body problem. Discrete and Continuous Dynamical Systems, 2017, 37 (1) : 229-256. doi: 10.3934/dcds.2017009
[11]	Samuel R. Kaplan, Ernesto A. Lacomba, Jaume Llibre. Symbolic dynamics of the elliptic rectilinear restricted 3--body problem. Discrete and Continuous Dynamical Systems - S, 2008, 1 (4) : 541-555. doi: 10.3934/dcdss.2008.1.541
[12]	Shingo Takeuchi. The basis property of generalized Jacobian elliptic functions. Communications on Pure and Applied Analysis, 2014, 13 (6) : 2675-2692. doi: 10.3934/cpaa.2014.13.2675
[13]	Hugo Beirão da Veiga. Elliptic boundary value problems in spaces of continuous functions. Discrete and Continuous Dynamical Systems - S, 2016, 9 (1) : 43-52. doi: 10.3934/dcdss.2016.9.43
[14]	Ying-Chieh Lin, Tsung-Fang Wu. On the semilinear fractional elliptic equations with singular weight functions. Discrete and Continuous Dynamical Systems - B, 2021, 26 (4) : 2067-2084. doi: 10.3934/dcdsb.2020325
[15]	Yimei Li, Jiguang Bao. Semilinear elliptic system with boundary singularity. Discrete and Continuous Dynamical Systems, 2020, 40 (4) : 2189-2212. doi: 10.3934/dcds.2020111
[16]	Giuseppe Maria Coclite, Helge Holden, Kenneth H. Karlsen. Wellposedness for a parabolic-elliptic system. Discrete and Continuous Dynamical Systems, 2005, 13 (3) : 659-682. doi: 10.3934/dcds.2005.13.659
[17]	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
[18]	Daniela De Silva, Fausto Ferrari, Sandro Salsa. On two phase free boundary problems governed by elliptic equations with distributed sources. Discrete and Continuous Dynamical Systems - S, 2014, 7 (4) : 673-693. doi: 10.3934/dcdss.2014.7.673
[19]	Daniela De Silva, Fausto Ferrari, Sandro Salsa. Recent progresses on elliptic two-phase free boundary problems. Discrete and Continuous Dynamical Systems, 2019, 39 (12) : 6961-6978. doi: 10.3934/dcds.2019239
[20]	Hua Chen, Wenbin Lv, Shaohua Wu. A free boundary problem for a class of parabolic-elliptic type chemotaxis model. Communications on Pure and Applied Analysis, 2018, 17 (6) : 2577-2592. doi: 10.3934/cpaa.2018122

2020 Impact Factor: 0.857

Tools

Metrics

Other articles
by authors

on AIMS
Francisco Crespo Sebastián Ferrer
on Google Scholar
Francisco Crespo Sebastián Ferrer

[Back to Top]