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

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

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

Sebastián Ferrer ^1, and Francisco Crespo ^1,

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

Received April 2014 Revised August 2014 Published December 2014

Abstract
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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.

Keywords: integrable system., Hopf fibration, quartic Hamiltonian.

Mathematics Subject Classification: Primary: 37N05, 53D20, 70E40; Secondary: 37J35, 70H06, 70H1.

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

References:

[1]	M. H. Andoyer, Cours de mécanique céleste, The Mathematical Gazette, 12 (1924), P30. doi: 10.2307/3603410.
[2]	F. Crespo, Hopf Fibration Reduction of a Quartic Model. An Application to Rotational and Orbital Dynamics, Ph.D thesis in preparation, Universidad de Murcia, 2014.
[3]	F. Crespo and S. Ferrer, On the extended Euler system and the Jacobi elliptic functions,, Submited to JGM., ().
[4]	R. H. Cushman and L. M. Bates, Global Aspects of Classical Integrable Systems, 2nd edition, Birkhäuser Verlag, Basel, 1997. doi: 10.1007/978-3-0348-8891-2.
[5]	R. H. Cushman and J. J. Duistermaat, A characterization of the Ligon-Schaaf regularization map, 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.
[6]	A. Deprit, The Lissajous transformation I. Basics, Celest. Mech., 51 (1991), 201-225. doi: 10.1007/BF00051691.
[7]	S. Ferrer, The Projective Andoyer transformation and the connection between the 4-D isotropic oscillator and Kepler systems,, , ().
[8]	T. Fukushima, Simple, regular, and efficient numerical integration of rotational motion, The Astronomical Journal, 135 (2008), 2298-2322. doi: 10.1088/0004-6256/135/6/2298.
[9]	G. Heckman and T. de Laat, On the regularization of the kepler problem, J. of Symplectic Geometry, 10 (2012), 463-473. doi: 10.4310/JSG.2012.v10.n3.a5.
[10]	D. D. Holm and J. E. Marsden, The rotor and the pendulum, 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.
[11]	H. Hopf, Über die Abbildungen der dreidimensionalen Sphäre auf die Kugelflsssche, Math. Ann., 104 (1931), 637-665. doi: 10.1007/BF01457962.
[12]	J. B. Kuipers, Quaternions and Rotation Sequences, Princeton university text, Princeton, New Jersey, 1999.
[13]	P. Kustaanheimo and E. Stiefel, Perturbation theory of Kepler motion based on spinor regularization, J. Reine Angew. Math., 218 (1965), 204-219. doi: 10.1515/crll.1965.218.204.
[14]	J. E. Marsden and T. S. Ratiu, Introduction to Mechanics and Symmetry. A Basic Exposition of Classical Mechanical Systems, 2nd edition, Springer, New York, 1999. doi: 10.1007/978-0-387-21792-5.
[15]	F. J. Molero, F. Crespo and S. Ferrer, Numerical integration versus analytical solution for a quartic Hamiltonian model in four dimensions,, In preparation., ().
[16]	S. Ferrer and J. Molero, Andoyer's variables and phases in the free rigid body, Journal of Geometric Mechanics, 6 (2014), 25-37. doi: 10.3934/jgm.2014.6.25.
[17]	J. Moser, Regularization of Kepler's problem and the averaging method on a manifold, Communication on pure and applied mathematics, 23 (1970), 609-636. doi: 10.1002/cpa.3160230406.
[18]	J. Moser and E. J. Zehnder, Notes on Dynamical Systems, Courant Lecture Notes in Mathematics, 12. New York University, Courant Institute of Mathematical Sciences, New York; American Mathematical Society, Providence, RI, 2005.
[19]	T. Ligon and M. Schaaf, On the global symmetry of the classical Kepler problem, Reports on Math. Phys., 9 (1976), 281-300. doi: 10.1016/0034-4877(76)90061-6.
[20]	J. P. Ortega and T. S. Ratiu, Momentum Maps and Hamiltonian Reduction, Birkhäuser Verlag, Basel, 2004. doi: 10.1007/978-1-4757-3811-7.
[21]	P. Saha, Interpreting the Kustaanheimo-Stiefel transform in gravitational dynamics, Mon. Not. R. Astron. Soc., 400 (2009), 228-231. doi: 10.1111/j.1365-2966.2009.15437.x.
[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.
[23]	J. Waldvogel, Quaternions and the perturbed Kepler problem, Celest. Mech. Dynamical Astron., 95 (2006), 201-212. doi: 10.1007/s10569-005-5663-7.
[24]	J. Waldvogel, Quaternions for regularizing Celestial Mechanics: The right way, Celest. Mech. Dynamical Astron., 102 (2008), 149-162. doi: 10.1007/s10569-008-9124-y.

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References:

[1]	M. H. Andoyer, Cours de mécanique céleste, The Mathematical Gazette, 12 (1924), P30. doi: 10.2307/3603410.
[2]	F. Crespo, Hopf Fibration Reduction of a Quartic Model. An Application to Rotational and Orbital Dynamics, Ph.D thesis in preparation, Universidad de Murcia, 2014.
[3]	F. Crespo and S. Ferrer, On the extended Euler system and the Jacobi elliptic functions,, Submited to JGM., ().
[4]	R. H. Cushman and L. M. Bates, Global Aspects of Classical Integrable Systems, 2nd edition, Birkhäuser Verlag, Basel, 1997. doi: 10.1007/978-3-0348-8891-2.
[5]	R. H. Cushman and J. J. Duistermaat, A characterization of the Ligon-Schaaf regularization map, 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.
[6]	A. Deprit, The Lissajous transformation I. Basics, Celest. Mech., 51 (1991), 201-225. doi: 10.1007/BF00051691.
[7]	S. Ferrer, The Projective Andoyer transformation and the connection between the 4-D isotropic oscillator and Kepler systems,, , ().
[8]	T. Fukushima, Simple, regular, and efficient numerical integration of rotational motion, The Astronomical Journal, 135 (2008), 2298-2322. doi: 10.1088/0004-6256/135/6/2298.
[9]	G. Heckman and T. de Laat, On the regularization of the kepler problem, J. of Symplectic Geometry, 10 (2012), 463-473. doi: 10.4310/JSG.2012.v10.n3.a5.
[10]	D. D. Holm and J. E. Marsden, The rotor and the pendulum, 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.
[11]	H. Hopf, Über die Abbildungen der dreidimensionalen Sphäre auf die Kugelflsssche, Math. Ann., 104 (1931), 637-665. doi: 10.1007/BF01457962.
[12]	J. B. Kuipers, Quaternions and Rotation Sequences, Princeton university text, Princeton, New Jersey, 1999.
[13]	P. Kustaanheimo and E. Stiefel, Perturbation theory of Kepler motion based on spinor regularization, J. Reine Angew. Math., 218 (1965), 204-219. doi: 10.1515/crll.1965.218.204.
[14]	J. E. Marsden and T. S. Ratiu, Introduction to Mechanics and Symmetry. A Basic Exposition of Classical Mechanical Systems, 2nd edition, Springer, New York, 1999. doi: 10.1007/978-0-387-21792-5.
[15]	F. J. Molero, F. Crespo and S. Ferrer, Numerical integration versus analytical solution for a quartic Hamiltonian model in four dimensions,, In preparation., ().
[16]	S. Ferrer and J. Molero, Andoyer's variables and phases in the free rigid body, Journal of Geometric Mechanics, 6 (2014), 25-37. doi: 10.3934/jgm.2014.6.25.
[17]	J. Moser, Regularization of Kepler's problem and the averaging method on a manifold, Communication on pure and applied mathematics, 23 (1970), 609-636. doi: 10.1002/cpa.3160230406.
[18]	J. Moser and E. J. Zehnder, Notes on Dynamical Systems, Courant Lecture Notes in Mathematics, 12. New York University, Courant Institute of Mathematical Sciences, New York; American Mathematical Society, Providence, RI, 2005.
[19]	T. Ligon and M. Schaaf, On the global symmetry of the classical Kepler problem, Reports on Math. Phys., 9 (1976), 281-300. doi: 10.1016/0034-4877(76)90061-6.
[20]	J. P. Ortega and T. S. Ratiu, Momentum Maps and Hamiltonian Reduction, Birkhäuser Verlag, Basel, 2004. doi: 10.1007/978-1-4757-3811-7.
[21]	P. Saha, Interpreting the Kustaanheimo-Stiefel transform in gravitational dynamics, Mon. Not. R. Astron. Soc., 400 (2009), 228-231. doi: 10.1111/j.1365-2966.2009.15437.x.
[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.
[23]	J. Waldvogel, Quaternions and the perturbed Kepler problem, Celest. Mech. Dynamical Astron., 95 (2006), 201-212. doi: 10.1007/s10569-005-5663-7.
[24]	J. Waldvogel, Quaternions for regularizing Celestial Mechanics: The right way, Celest. Mech. Dynamical Astron., 102 (2008), 149-162. doi: 10.1007/s10569-008-9124-y.

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