# American Institute of Mathematical Sciences

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

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

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
##### References:
 [1] M. H. Andoyer, Cours de mécanique céleste,, The Mathematical Gazette, 12 (1924). 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, (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, (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. 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. 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. 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. 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, 99 (1991), 189. Google Scholar [11] H. Hopf, Über die Abbildungen der dreidimensionalen Sphäre auf die Kugelflsssche,, Math. Ann., 104 (1931), 637. doi: 10.1007/BF01457962. Google Scholar [12] J. B. Kuipers, Quaternions and Rotation Sequences,, Princeton university text, (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. 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, (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. 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. doi: 10.1002/cpa.3160230406. Google Scholar [18] J. Moser and E. J. Zehnder, Notes on Dynamical Systems,, Courant Lecture Notes in Mathematics, (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. 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, (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. 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. 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. doi: 10.1007/s10569-008-9124-y. Google Scholar

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##### References:
 [1] M. H. Andoyer, Cours de mécanique céleste,, The Mathematical Gazette, 12 (1924). 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, (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, (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. 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. 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. 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. 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, 99 (1991), 189. Google Scholar [11] H. Hopf, Über die Abbildungen der dreidimensionalen Sphäre auf die Kugelflsssche,, Math. Ann., 104 (1931), 637. doi: 10.1007/BF01457962. Google Scholar [12] J. B. Kuipers, Quaternions and Rotation Sequences,, Princeton university text, (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. 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, (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. 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. doi: 10.1002/cpa.3160230406. Google Scholar [18] J. Moser and E. J. Zehnder, Notes on Dynamical Systems,, Courant Lecture Notes in Mathematics, (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. 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, (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. 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. 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. doi: 10.1007/s10569-008-9124-y. Google Scholar
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