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Parametric quartic Hamiltonian model. A unified treatment of classic integrable systems
1. | Departamento de Matemática Aplicada, Universidad de Murcia, 30100 Espinardo, Spain |
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). 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. |
[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,, , (). 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. |
[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., (). 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. |
[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. 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. |
[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). 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. |
[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,, , (). 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. |
[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., (). 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. |
[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. 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. |
[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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