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
References:
[1]

M. H. Andoyer, Cours de mécanique céleste,, The Mathematical Gazette, 12 (1924). 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, (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, (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. 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. 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. 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. 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, 99 (1991), 189.

[11]

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

[12]

J. B. Kuipers, Quaternions and Rotation Sequences,, Princeton university text, (1999).

[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.

[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.

[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. 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. doi: 10.1002/cpa.3160230406.

[18]

J. Moser and E. J. Zehnder, Notes on Dynamical Systems,, Courant Lecture Notes in Mathematics, (2005).

[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.

[20]

J. P. Ortega and T. S. Ratiu, Momentum Maps and Hamiltonian Reduction,, Birkhäuser Verlag, (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. 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. 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. doi: 10.1007/s10569-008-9124-y.

show all references

References:
[1]

M. H. Andoyer, Cours de mécanique céleste,, The Mathematical Gazette, 12 (1924). 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, (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, (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. 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. 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. 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. 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, 99 (1991), 189.

[11]

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

[12]

J. B. Kuipers, Quaternions and Rotation Sequences,, Princeton university text, (1999).

[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.

[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.

[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. 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. doi: 10.1002/cpa.3160230406.

[18]

J. Moser and E. J. Zehnder, Notes on Dynamical Systems,, Courant Lecture Notes in Mathematics, (2005).

[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.

[20]

J. P. Ortega and T. S. Ratiu, Momentum Maps and Hamiltonian Reduction,, Birkhäuser Verlag, (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. 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. 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. doi: 10.1007/s10569-008-9124-y.

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