Perturbed $(2n-1)$-Dimensional Kepler Problem and the Nilpotent Adjoint Orbits of $U(n, n)$

Anatol Odzijewicz

Symmetry, integrability and geometry: methods and applications, Tome 16 (2020)

Résumé

We study the regularized $(2n-1)$-Kepler problem and other Hamiltonian systems which are related to the nilpotent coadjoint orbits of $U(n,n)$. The Kustaanheimo–Stiefel and Cayley regularization procedures are discussed and their equivalence is shown. Some integrable generalization (perturbation) of $(2n-1)$-Kepler problem is proposed.

Détail

Keywords: integrable Hamiltonian systems, Kepler problem, nonlinear differential equations, symplectic geometry, Poisson geometry, Kustaanheimo–Stiefel transformation, celestial mechanics.

Bibliographie

[1] Barut A. O., Kleinert H., “Transition probabilities of the hydrogen atom from noncompact dynamical groups”, Phys. Rev., 156 (1967), 1541–1545 | DOI

[2] Bouarroudj S., Meng G., “The classical dynamic symmetry for the ${\rm U}(1)$-Kepler problems”, J. Geom. Phys., 124 (2018), 1–15 | DOI | MR | Zbl

[3] Cordani B., “On the generalisation of the Kustaanheimo-Stiefel transformations”, J. Phys. A: Math. Gen., 22 (1989), 2441–2446 | DOI | MR | Zbl

[4] Cordani B., Reina C., “Spinor regularization of the $n$-dimensional Kepler problem”, Lett. Math. Phys., 13 (1987), 79–82 | DOI | MR

[5] Efstathiou K., Sadovskii D. A., “Normalization and global analysis of perturbations of the hydrogen atom”, Rev. Mod. Phys., 82 (2010), 2099–2154 | DOI

[6] Faraut J., Korányi A., Analysis on symmetric cones, Oxford Mathematical Monographs, The Clarendon Press, Oxford University Press, New York, 1994 | MR

[7] Ferrer S., Crespo F., “Alternative angle-based approach to the ${\mathcal{KS}}$-map. An interpretation through symmetry and reduction”, J. Geom. Mech., 10 (2018), 359–372, arXiv: 1711.08530 | DOI | MR | Zbl

[8] Horowski M., Odzijewicz A., “Geometry of the Kepler system in coherent states approach”, Ann. Inst. H. Poincaré Phys. Théor., 59 (1993), 69–89 | MR | Zbl

[9] Iwai T., “The geometry of the ${\rm SU}(2)$ Kepler problem”, J. Geom. Phys., 7 (1990), 507–535 | DOI | MR | Zbl

[10] Iwai T., “A dynamical group ${\rm SU}(2,2)$ and its use in the MIC-Kepler problem”, J. Phys. A: Math. Gen., 26 (1993), 609–630 | DOI | MR | Zbl

[11] Iwai T., Matsumoto S., “Poisson mechanics for perturbed MIC-Kepler problems at both positive and negative energies”, J. Phys. A: Math. Theor., 45 (2012), 365203, 34 pp. | DOI | MR | Zbl

[12] Iwai T., Uwano Y., “The four-dimensional conformal Kepler problem reduces to the three-dimensional Kepler problem with a centrifugal potential and Dirac's monopole field. Classical theory”, J. Math. Phys., 27 (1986), 1523–1529 | DOI | MR | Zbl

[13] Kirillov A. A., Elements of the theory of representations, Grundlehren der Mathematischen Wissenschaften, 220, Springer-Verlag, Berlin–New York, 1976 | DOI | MR | Zbl

[14] Kummer M., “On the regularization of the Kepler problem”, Comm. Math. Phys., 84 (1982), 133–152 | DOI | MR | Zbl

[15] Kustaanheimo P., Stiefel E., “Perturbation theory of Kepler motion based on spinor regularization”, J. Reine Angew. Math., 218 (1965), 204–219 | DOI | MR | Zbl

[16] Lang S., Algebra, Graduate Texts in Mathematics, 211, 3rd ed., Springer-Verlag, New York, 2002 | DOI | MR | Zbl

[17] Malkin I. A., Man'ko V. I., “Symmetry of the hydrogen atom”, JETP Lett., 2 (1965), 146–148 | MR

[18] McIntosh H. V., Cisneros A., “Degeneracy in the presence of a magnetic monopole”, J. Math. Phys., 11 (1970), 896–916 | DOI | MR

[19] Meng G., “Generalized MICZ-Kepler problems and unitary highest weight modules, II”, J. Lond. Math. Soc., 81 (2010), 663–678, arXiv: 0704.2936 | DOI | MR | Zbl

[20] Meng G., “Euclidean Jordan algebras, hidden actions, and J-Kepler problems”, J. Math. Phys., 52 (2011), 112104, 35 pp., arXiv: 0911.2977 | DOI | MR | Zbl

[21] Meng G., “Generalized Kepler problems. I Without magnetic charges”, J. Math. Phys., 54 (2013), 012109, 25 pp., arXiv: 1104.2585 | DOI | MR | Zbl

[22] Meng G., “The universal Kepler problem”, J. Geom. Symmetry Phys., 36 (2014), 47–57, arXiv: 1011.6609 | DOI | MR | Zbl

[23] Meng G., “On the trajectories of ${\rm U}(1)$-Kepler problems”, Geometry, Integrability and Quantization XVI, Avangard Prima, Sofia, 2015, 219–230 | DOI | MR | Zbl

[24] Mladenov I., Tsanov V., “Geometric quantization of the multidimensional Kepler problem”, J. Geom. Phys., 2 (1985), 17–24 | DOI | MR | Zbl

[25] Mladenov I. M., Tsanov V. V., “Geometric quantisation of the MIC-Kepler problem”, J. Phys. A: Math. Gen., 20 (1987), 5865–5871 | DOI | MR | Zbl

[26] Moser J., “Regularization of Kepler's problem and the averaging method on a manifold”, Comm. Pure Appl. Math., 23 (1970), 609–636 | DOI | MR | Zbl

[27] Odzijewicz A., “A conformal holomorphic field theory”, Comm. Math. Phys., 107 (1986), 561–575 | DOI | MR | Zbl

[28] Odzijewicz A., Świȩtochowski M., “Coherent states map for MIC-Kepler system”, J. Math. Phys., 38 (1997), 5010–5030 | DOI | MR | Zbl

[29] Odzijewicz A., Wawreniuk E., “Classical and quantum Kummer shape algebras”, J. Phys. A: Math. Theor., 49 (2016), 265202, 33 pp., arXiv: 1512.09279 | DOI | MR | Zbl

[30] Penrose R., “Twistor algebra”, J. Math. Phys., 8 (1967), 345–366 | DOI | MR | Zbl

[31] Simms D. J., “Bohr–Sommerfeld orbits and quantizable symplectic manifolds”, Proc. Cambridge Philos. Soc., 73 (1973), 489–491 | DOI | MR | Zbl

[32] Stiefel E. L., Scheifele G., Linear and regular celestial mechanics, Grundlehren der mathematischen Wissenschaften, 174, Springer-Verlag, Berlin–Heidelberg, 1971 | Zbl

[33] Zhao L., “Kustaanheimo–Stiefel regularization and the quadrupolar conjugacy”, Regul. Chaotic Dyn., 20 (2015), 19–36 | DOI | MR | Zbl

[34] Zwanzinger D., “Exactly soluble nonrelativistic model of particles with both electric and magnetic charges”, Phys. Rev., 176 (1968), 1480–1488 | DOI