December  2014, 6(4): 479-502. doi: 10.3934/jgm.2014.6.479

Parametric quartic Hamiltonian model. A unified treatment of classic integrable systems

1. 

Departamento de Matemática Aplicada, Universidad de Murcia, 30100 Espinardo, Spain

Received  April 2014 Revised  August 2014 Published  December 2014

Related to the components of the quaternionic Hopf mapping, we propose a parametric Hamiltonian function in $\mathbb{T}^*\mathbb{R}^4$ which is a homogeneous quartic polynomial with six parameters, defining an integrable family of Hamiltonian systems. The key feature of the model is its nested Hamiltonian-Poisson structure, which appears as two extended Euler systems in the reduced equations. This is fully exploited in the process of integration, where we find two 1-DOF subsystems and a quadrature involving both of them. The solution is quasi-periodic, expressed by means of Jacobi elliptic functions and integrals, based on two periods. For a suitable choice of the parameters, some remarkable classical models such as the Kepler, geodesic flow, isotropic oscillator and free rigid body systems appear as particular cases.
Citation: Sebastián Ferrer, Francisco Crespo. Parametric quartic Hamiltonian model. A unified treatment of classic integrable systems. Journal of Geometric Mechanics, 2014, 6 (4) : 479-502. doi: 10.3934/jgm.2014.6.479
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F. J. Molero, F. Crespo and S. Ferrer, Numerical integration versus analytical solution for a quartic Hamiltonian model in four dimensions,, In preparation., ().   Google Scholar

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Journal of Geometric Mechanics, 6 (2014), 25-37. doi: 10.3934/jgm.2014.6.25.  Google Scholar

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Courant Lecture Notes in Mathematics, 12. New York University, Courant Institute of Mathematical Sciences, New York; American Mathematical Society, Providence, RI, 2005.  Google Scholar

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Birkhäuser Verlag, Basel, 2004. doi: 10.1007/978-1-4757-3811-7.  Google Scholar

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J. C. van der Meer, F. Crespo and S. Ferrer, Generalized Hopf fibration and geometric $SO(3)$ reduction of the 4-DOF harmonic oscillator,, , (): 14.   Google Scholar

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show all references

References:
[1]

The Mathematical Gazette, 12 (1924), P30. doi: 10.2307/3603410.  Google Scholar

[2]

Ph.D thesis in preparation, Universidad de Murcia, 2014. Google Scholar

[3]

F. Crespo and S. Ferrer, On the extended Euler system and the Jacobi elliptic functions,, Submited to JGM., ().   Google Scholar

[4]

2nd edition, Birkhäuser Verlag, Basel, 1997. doi: 10.1007/978-3-0348-8891-2.  Google Scholar

[5]

Comm. Pure and Appl. Math., 50 (1997), 773-787. doi: 10.1002/(SICI)1097-0312(199708)50:8<773::AID-CPA3>3.0.CO;2-3.  Google Scholar

[6]

Celest. Mech., 51 (1991), 201-225. doi: 10.1007/BF00051691.  Google Scholar

[7]

S. Ferrer, The Projective Andoyer transformation and the connection between the 4-D isotropic oscillator and Kepler systems,, , ().   Google Scholar

[8]

The Astronomical Journal, 135 (2008), 2298-2322. doi: 10.1088/0004-6256/135/6/2298.  Google Scholar

[9]

J. of Symplectic Geometry, 10 (2012), 463-473. doi: 10.4310/JSG.2012.v10.n3.a5.  Google Scholar

[10]

In Symplectic Geometry and Mathematical Physics, Actes du colloque en l'honneur de Jean-Marie Souriau, Ed by P. Donato et al., Prog. in Math., Birkhäuser Verlag, Basel, 99 (1991), 189-203.  Google Scholar

[11]

Math. Ann., 104 (1931), 637-665. doi: 10.1007/BF01457962.  Google Scholar

[12]

Princeton university text, Princeton, New Jersey, 1999.  Google Scholar

[13]

J. Reine Angew. Math., 218 (1965), 204-219. doi: 10.1515/crll.1965.218.204.  Google Scholar

[14]

2nd edition, Springer, New York, 1999. doi: 10.1007/978-0-387-21792-5.  Google Scholar

[15]

F. J. Molero, F. Crespo and S. Ferrer, Numerical integration versus analytical solution for a quartic Hamiltonian model in four dimensions,, In preparation., ().   Google Scholar

[16]

Journal of Geometric Mechanics, 6 (2014), 25-37. doi: 10.3934/jgm.2014.6.25.  Google Scholar

[17]

Communication on pure and applied mathematics, 23 (1970), 609-636. doi: 10.1002/cpa.3160230406.  Google Scholar

[18]

Courant Lecture Notes in Mathematics, 12. New York University, Courant Institute of Mathematical Sciences, New York; American Mathematical Society, Providence, RI, 2005.  Google Scholar

[19]

Reports on Math. Phys., 9 (1976), 281-300. doi: 10.1016/0034-4877(76)90061-6.  Google Scholar

[20]

Birkhäuser Verlag, Basel, 2004. doi: 10.1007/978-1-4757-3811-7.  Google Scholar

[21]

Mon. Not. R. Astron. Soc., 400 (2009), 228-231. doi: 10.1111/j.1365-2966.2009.15437.x.  Google Scholar

[22]

J. C. van der Meer, F. Crespo and S. Ferrer, Generalized Hopf fibration and geometric $SO(3)$ reduction of the 4-DOF harmonic oscillator,, , (): 14.   Google Scholar

[23]

Celest. Mech. Dynamical Astron., 95 (2006), 201-212. doi: 10.1007/s10569-005-5663-7.  Google Scholar

[24]

Celest. Mech. Dynamical Astron., 102 (2008), 149-162. doi: 10.1007/s10569-008-9124-y.  Google Scholar

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