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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

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.  Google Scholar

[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. Google Scholar

[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. Google Scholar

[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.  Google Scholar

[5]

A. G. Greenhill, The Applications of Elliptic Functions, Macmillan and Co., London, 1892. Google Scholar

[6]

C. Gudermann, Theorie der Modular Functionen,, Crelle's Journal, ().   Google Scholar

[7]

J. K. Hale, Ordinary Differential Equations, Wiley-Interscience, New York, 1969.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[10]

D. F. Lawden, Elliptic Functions and Aplications, Vol. 80, Springer-Verlag, New York, 1989. doi: 10.1007/978-1-4757-3980-0.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[14]

Y. Nambu, Generalized Hamiltonian mechanics, Phys. Rev., 7 (1973), 2405-2412. doi: 10.1103/PhysRevD.7.2405.  Google Scholar

[15]

M. Puta, On the dynamics of the rigid body with two torques, C. R. Acad. Sci. Paris, 317 (1993), 377-380.  Google Scholar

[16]

M. Puta, Stability and control in spacecraft dynamics, Journal of Lie Theory, 7 (1997), 269-278.  Google Scholar

[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.  Google Scholar

[18]

A. Weinstein, The local structure of Poisson manifolds, J. Differential Geom., 18 (1983), 523-557.  Google Scholar

[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.  Google Scholar

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.  Google Scholar

[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. Google Scholar

[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. Google Scholar

[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.  Google Scholar

[5]

A. G. Greenhill, The Applications of Elliptic Functions, Macmillan and Co., London, 1892. Google Scholar

[6]

C. Gudermann, Theorie der Modular Functionen,, Crelle's Journal, ().   Google Scholar

[7]

J. K. Hale, Ordinary Differential Equations, Wiley-Interscience, New York, 1969.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[10]

D. F. Lawden, Elliptic Functions and Aplications, Vol. 80, Springer-Verlag, New York, 1989. doi: 10.1007/978-1-4757-3980-0.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[14]

Y. Nambu, Generalized Hamiltonian mechanics, Phys. Rev., 7 (1973), 2405-2412. doi: 10.1103/PhysRevD.7.2405.  Google Scholar

[15]

M. Puta, On the dynamics of the rigid body with two torques, C. R. Acad. Sci. Paris, 317 (1993), 377-380.  Google Scholar

[16]

M. Puta, Stability and control in spacecraft dynamics, Journal of Lie Theory, 7 (1997), 269-278.  Google Scholar

[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.  Google Scholar

[18]

A. Weinstein, The local structure of Poisson manifolds, J. Differential Geom., 18 (1983), 523-557.  Google Scholar

[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.  Google Scholar

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