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On the extended Euler system and the Jacobi and Weierstrass elliptic functions
1. | Departamento de Matemática Aplicada, Universidad de Murcia, 30071 Espinardo, Spain, Spain |
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. |
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