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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

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.  Google Scholar

[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. Google Scholar

[3]

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

[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.  Google Scholar

[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.  Google Scholar

[6]

A. Deprit, The Lissajous transformation I. Basics, 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]

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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[11]

H. Hopf, Über die Abbildungen der dreidimensionalen Sphäre auf die Kugelflsssche, Math. Ann., 104 (1931), 637-665. doi: 10.1007/BF01457962.  Google Scholar

[12]

J. B. Kuipers, Quaternions and Rotation Sequences, Princeton university text, Princeton, New Jersey, 1999.  Google Scholar

[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.  Google Scholar

[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.  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]

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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  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]

J. Waldvogel, Quaternions and the perturbed Kepler problem, Celest. Mech. Dynamical Astron., 95 (2006), 201-212. doi: 10.1007/s10569-005-5663-7.  Google Scholar

[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.  Google Scholar

show all references

References:
[1]

M. H. Andoyer, Cours de mécanique céleste, The Mathematical Gazette, 12 (1924), P30. doi: 10.2307/3603410.  Google Scholar

[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. Google Scholar

[3]

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

[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.  Google Scholar

[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.  Google Scholar

[6]

A. Deprit, The Lissajous transformation I. Basics, 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]

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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[11]

H. Hopf, Über die Abbildungen der dreidimensionalen Sphäre auf die Kugelflsssche, Math. Ann., 104 (1931), 637-665. doi: 10.1007/BF01457962.  Google Scholar

[12]

J. B. Kuipers, Quaternions and Rotation Sequences, Princeton university text, Princeton, New Jersey, 1999.  Google Scholar

[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.  Google Scholar

[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.  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]

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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  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]

J. Waldvogel, Quaternions and the perturbed Kepler problem, Celest. Mech. Dynamical Astron., 95 (2006), 201-212. doi: 10.1007/s10569-005-5663-7.  Google Scholar

[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.  Google Scholar

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