Geodesic
Quasi-Bi-Hamiltonian Structures of the 2-Dimensional Kepler Problem
Symmetry, integrability and geometry: methods and applications, Tome 12 (2016) Cet article a éte moissonné depuis la source Math-Net.Ru

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The existence of quasi-bi-Hamiltonian structures for the Kepler problem is studied. We first relate the superintegrability of the system with the existence of two complex functions endowed with very interesting Poisson bracket properties and then we prove the existence of a quasi-bi-Hamiltonian structure by making use of these two functions. The paper can be considered as divided in two parts. In the first part a quasi-bi-Hamiltonian structure is obtained by making use of polar coordinates and in the second part a new quasi-bi-Hamiltonian structure is obtained by making use of the separability of the system in parabolic coordinates.
Keywords: Kepler problem; superintegrability; complex structures; bi-Hamiltonian structures; quasi-bi-Hamiltonian structures. 
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     title = {Quasi-Bi-Hamiltonian {Structures} of the {2-Dimensional} {Kepler} {Problem}},
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Jose F. Cariñena; Manuel F. Rañada. Quasi-Bi-Hamiltonian Structures of the 2-Dimensional Kepler Problem. Symmetry, integrability and geometry: methods and applications, Tome 12 (2016). http://geodesic.mathdoc.fr/item/SIGMA_2016_12_a9/

[1] Błaszak M., “Bi-Hamiltonian representation of Stäckel systems”, Phys. Rev. E, 79 (2009), 056607, 9 pp., arXiv: 0904.2070 | DOI | MR

[2] Boualem H., Brouzet R., Rakotondralambo J., Quasi-bi-Hamiltonian systems: why the Pfaffian case?, Phys. Lett. A, 359 (2006), 559–563 | DOI | MR | Zbl

[3] Boualem H., Brouzet R., Rakotondralambo J., “About the separability of completely integrable quasi-bi-Hamiltonian systems with compact levels”, Differential Geom. Appl., 26 (2008), 583–591 | DOI | MR | Zbl

[4] Brouzet R., Caboz R., Rabenivo J., Ravoson V., “Two degrees of freedom quasi-bi-Hamiltonian systems”, J. Phys. A: Math. Gen., 29 (1996), 2069–2076 | DOI | MR | Zbl

[5] Cariñena J. F., Guha P., Rañada M. F., “Quasi-Hamiltonian structure and Hojman construction”, J. Math. Anal. Appl., 332 (2007), 975–988 | DOI | MR | Zbl

[6] Cariñena J. F., Guha P., Rañada M. F., “Hamiltonian and quasi-Hamiltonian systems, Nambu–Poisson structures and symmetries”, J. Phys. A: Math. Theor., 41 (2008), 335209, 11 pp. | DOI | MR | Zbl

[7] Cariñena J. F., Ibort L. A., “Non-Noether constants of motion”, J. Phys. A: Math. Gen., 16 (1983), 1–7 | DOI | MR | Zbl

[8] Cariñena J. F., Marmo G., Rañada M. F., “Non-symplectic symmetries and bi-Hamiltonian structures of the rational harmonic oscillator”, J. Phys. A: Math. Gen., 35 (2002), L679–L686, arXiv: hep-th/0210260 | DOI | MR | Zbl

[9] Cariñena J. F., Rañada M. F., “Canonoid transformations from a geometric perspective”, J. Math. Phys., 29 (1988), 2181–2186 | DOI | MR

[10] Cariñena J. F., Rañada M. F., “Generating functions, bi-Hamiltonian systems, and the quadratic-Hamiltonian theorem”, J. Math. Phys., 31 (1990), 801–807 | DOI | MR | Zbl

[11] Cariñena J. F., Rañada M. F., Santander M., “Two important examples of nonlinear oscillators”, Proceedings of 10th International Conference in Modern Group Analysis (Larnaca, Cyprus, 2004), eds. N. H. Ibragimov, C. Sophocleous, P. A. Damianou, 2005, 39–46, arXiv: math-ph/0505028 | MR

[12] Crampin M., Marmo G., Rubano C., “Conditions for the complete integrability of a dynamical system admitting alternative Lagrangians”, Phys. Lett. A, 97 (1983), 88–90 | DOI | MR

[13] Crampin M., Prince G. E., “Alternative Lagrangians for spherically symmetric potentials”, J. Math. Phys., 29 (1988), 1551–1554 | DOI | MR | Zbl

[14] Crampin M., Sarlet W., “A class of nonconservative Lagrangian systems on Riemannian manifolds”, J. Math. Phys., 42 (2001), 4313–4326 | DOI | MR | Zbl

[15] Crampin M., Sarlet W., “Bi-quasi-Hamiltonian systems”, J. Math. Phys., 43 (2002), 2505–2517 | DOI | MR | Zbl

[16] Crampin M., Sarlet W., Thompson G., “Bi-differential calculi and bi-Hamiltonian systems”, J. Phys. A: Math. Gen., 33 (2000), L177–L180 | DOI | MR | Zbl

[17] De Filippo S., Vilasi G., Marmo G., Salerno M., “A new characterization of completely integrable systems”, Nuovo Cimento B, 83 (1984), 97–112 | DOI | MR

[18] Fernandes R. L., “Completely integrable bi-Hamiltonian systems”, J. Dynam. Differential Equations, 6 (1994), 53–69 | DOI | MR | Zbl

[19] Grigoryev Y. A., Tsiganov A. V., “On bi-Hamiltonian formulation of the perturbed Kepler problem”, J. Phys. A: Math. Theor., 48 (2015), 175206, 7 pp., arXiv: 1504.02211 | DOI | MR | Zbl

[20] Henneaux M., Shepley L. C., “Lagrangians for spherically symmetric potentials”, J. Math. Phys., 23 (1982), 2101–2107 | DOI | MR | Zbl

[21] Marmo G., “A geometrical characterization of completely integrable systems”, Proceedings of the International Meeting on Geometry and Physics (Florence, 1982), Pitagora, Bologna, 1983, 257–262 | MR

[22] McLenaghan R. G., Smirnov R. G., “A class of Liouville-integrable Hamiltonian systems with two degrees of freedom”, J. Math. Phys., 41 (2000), 6879–6889 | DOI | MR | Zbl

[23] Morosi C., Tondo G., “Quasi-bi-Hamiltonian systems and separability”, J. Phys. A: Math. Gen., 30 (1997), 2799–2806, arXiv: solv-int/9702006 | DOI | MR | Zbl

[24] Morosi C., Tondo G., “On a class of dynamical systems both quasi-bi-Hamiltonian and bi-Hamiltonian”, Phys. Lett. A, 247 (1998), 59–64, arXiv: solv-int/9811007 | DOI | MR | Zbl

[25] Rakotondralambo J., “A cohomological obstruction for global quasi-bi-Hamiltonian fields”, Phys. Lett. A, 375 (2011), 1064–1068 | DOI | MR | Zbl

[26] Rañada M. F., “Alternative Lagrangians for central potentials”, J. Math. Phys., 32 (1991), 2764–2769 | DOI | MR

[27] Rañada M. F., “Dynamical symmetries, bi-Hamiltonian structures, and superintegrable $n=2$ systems”, J. Math. Phys., 41 (2000), 2121–2134 | DOI | MR

[28] Rañada M. F., “A system of $n=3$ coupled oscillators with magnetic terms: symmetries and integrals of motion”, SIGMA, 1 (2005), 004, 7 pp., arXiv: nlin.SI/0511007 | DOI | MR

[29] Rañada M. F., “A new approach to the higher order superintegrability of the Tremblay–Turbiner–Winternitz system”, J. Phys. A: Math. Theor., 45 (2012), 465203, 9 pp., arXiv: 1211.2919 | DOI | MR

[30] Rañada M. F., “Higher order superintegrability of separable potentials with a new approach to the Post–Winternitz system”, J. Phys. A: Math. Theor., 46 (2013), 125206, 9 pp., arXiv: 1303.4877 | DOI | MR

[31] Rañada M. F., “The Tremblay–Turbiner–Winternitz system on spherical and hyperbolic spaces: superintegrability, curvature-dependent formalism and complex factorization”, J. Phys. A: Math. Theor., 47 (2014), 165203, 9 pp., arXiv: 1403.6266 | DOI | MR

[32] Rañada M. F., “The Post–Winternitz system on spherical and hyperbolic spaces: a proof of the superintegrability making use of complex functions and a curvature-dependent formalism”, Phys. Lett. A, 379 (2015), 2267–2271 | DOI | MR

[33] Rañada M. F., Rodríguez M. A., Santander M., “A new proof of the higher-order superintegrability of a noncentral oscillator with inversely quadratic nonlinearities”, J. Math. Phys., 51 (2010), 042901, 11 pp., arXiv: 1002.3870 | DOI | MR

[34] Rauch-Wojciechowski S., “A bi-Hamiltonian formulation for separable potentials and its application to the Kepler problem and the Euler problem of two centers of gravitation”, Phys. Lett. A, 160 (1991), 149–154 | DOI | MR

[35] Tempesta P., Tondo G., “Generalized Lenard chains, separation of variables, and superintegrability”, Phys. Rev. E, 85 (2012), 046602, 11 pp., arXiv: 1205.6937 | DOI

[36] Zeng Y. B., Ma W. X., “Families of quasi-bi-Hamiltonian systems and separability”, J. Math. Phys., 40 (1999), 4452–4473, arXiv: solv-int/9906005 | DOI | MR | Zbl