Curvature as an Integrable Deformation

Referencias bibliográficas

• A.M. Perelomov, Integrable Systems of Classical Mechanics and Lie Algebras (Birkhäuser, Berlin, 1990)
• A. Goriely, Integrability and Nonintegrability of Dynamical Systems (World Scientific, Singapore, 2001)
• T.G. Vozmischeva, Integrable Problems of Celestial Mechanics in Spaces of Constant Curvature. Astrophysics and Space Science Library, vol. 295 (Kluwer, Dordrecht, 2003)
• D. Boccaletti, G. Pucacco, Theory of Orbits (Springer, Berlin, 2004)
• W. Miller, Jr., S. Post, P. Winternitz, Classical and quantum superintegrability with applications. J. Phys. A Math. Theor. 46, 423001 (2013). https://doi.org/10.1088/1751-8113/46/42/423001
• A. Ballesteros, F.J. Herranz, M.A. del Olmo, M. Santander, Quantum structure of the motion groups of the two-dimensional Cayley–Klein geometries. J. Phys. A: Math. Gen. 26, 5801–5823 (1993). https://doi.org/10.1088/0305-4470/26/21/019
• M.F. Rañada, M. Santander, Superintegrable systems on the two-dimensional sphere S 2 and the hyperbolic plane H 2. J. Math. Phys. 40, 5026–5057 (1999). https://doi.org/10.1063/1.533014
• F.J. Herranz, R. Ortega, M. Santander, Trigonometry of spacetimes: a new self-dual approach to a curvature/signature (in)dependent trigonometry. J. Phys. A: Math. Gen. 33, 4525–4551 (2000). https://doi.org/10.1088/0305-4470/33/24/309
• F.J. Herranz, M. Santander, Conformal symmetries of spacetimes. J. Phys. A: Math. Gen. 35, 6601–6618 (2002). https://doi.org/10.1088/0305-4470/35/31/306
• M.F. Rañada, M. Santander, On some properties of harmonic oscillator on spaces of constant curvature. Rep. Math. Phys. 49, 335–343 (2002). https://doi.org/10.1016/S0034-4877(02)80031-3
• M.F. Rañada, M. Santander, On harmonic oscillators on the two-dimensional sphere S 2 and the hyperbolic plane H 2. J. Math. Phys. 43, 431–451 (2002). https://doi.org/10.1063/1.1423402
• A. Ballesteros, F.J. Herranz, M. Santander, T. Sanz-Gil, Maximal superintegrability on N-dimensional curved spaces. J. Phys. A: Math. Gen. 36, L93–L99 (2003). https://doi.org/10.1088/0305-4470/36/7/101
• F.J. Herranz, A. Ballesteros, M. Santander, T. Sanz-Gil, Maximally superintegrable Smorodinsky–Winternitz systems on the N-dimensional sphere and hyperbolic spaces, in Superintegrability in Classical and Quantum Systems, ed. by P. Tempesta et al. CRM Proceedings and Lecture Notes, vol. 37 (American Mathematical Society, Providence, 2004), pp. 75–89. https://doi.org/10.1090/crmp/037
• F.J. Herranz, A. Ballesteros, Superintegrability on three-dimensional Riemannian and relativistic spaces of constant curvature. Symmetry Integrability Geom. Methods Appl. 2, 010 (2006). https://doi.org/10.3842/SIGMA.2006.010
• A. Ballesteros, F.J. Herranz, Universal integrals for superintegrable systems on N-dimensional spaces of constant curvature. J. Phys. A: Math. Theor. 40, F51–F59 (2007). https://doi.org/10.1088/1751-8113/40/2/F01
• J.F. Cariñena, M.F. Rañada, M. Santander, The quantum harmonic oscillator on the sphere and the hyperbolic plane. Ann. Phys. 322, 2249–2278 (2007). https://doi.org/10.1016/j.aop.2006.10.010
• J.F. Cariñena, M.F. Rañada, M. Santander, Superintegrability on curved spaces, orbits and momentum hodographs: revisiting a classical result by Hamilton. J. Phys. A: Math. Theor. 40, 13645–13666 (2007). https://doi.org/10.1088/1751-8113/40/45/010
• J.F. Cariñena, M.F. Rañada, M. Santander, The Kepler problem and the Laplace-Runge-Lenz vector on spaces of constant curvature and arbitrary signature. Qual. Theory Dyn. Syst. 7, 87–99 (2008). https://doi.org/10.1007/s12346-008-0004-3
• A. Ballesteros, F.J. Herranz, Maximal superintegrability of the generalized Kepler–Coulomb system on N-dimensional curved spaces. J. Phys. A: Math. Theor. 42, 245203 (2009). https://doi.org/10.1088/1751-8113/42/24/245203
• F. Diacu, E. Pérez-Chavela, Homographic solutions of the curved 3-body problem. J. Differ. Equ. 250, 340–366 (2011). https://doi.org/10.1016/j.jde.2010.08.011
• F. Diacu, E. Pérez-Chavela, M. Santoprete, The n-body problem in spaces of constant curvature. Part I: relative equilibria. J. Nonlinear Sci. 22, 247–266 (2012). https://doi.org/10.1007/s00332-011-9116-z
• F. Diacu, E. Pérez-Chavela, M. Santoprete, The n-body problem in spaces of constant curvature. Part II: singularities equilibria. J. Nonlinear Sci. 22, 267–275 (2012). https://doi.org/10.1007/s00332-011-9117-y
• F. Diacu, Relative equilibria in the 3-dimensional curved n-body problem. Memoirs Am. Math. Soc. 228, 1071 (2014). http://dx.doi.org/10.1090/memo/1071
• C. Gonera, M. Kaszubska, Superintegrable systems on spaces of constant curvature. Ann. Phys. 364, 91–102 (2014). https://doi.org/10.1016/j.aop.2014.04.005
• M.F. Rañada, The Tremblay-Turbiner-Winternitz system on spherical and hyperbolic spaces: superintegrability, curvature-dependent formalism and complex factorization. J. Phys. A: Math. Theor. 47, 165203 (2014). https://doi.org/10.1088/1751-8113/47/16/165203
• 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, 2267–2271 (2015). https://doi.org/10.1016/j.physleta.2015.07.043
• M.F. Rañada, Superintegrable deformations of superintegrable systems: quadratic superintegrability and higher-order superintegrability. J. Math. Phys. 56, 042703 (2015). https://doi.org/10.1063/1.4918611
• C.M. Chanu, L. Degiovanni, G. Rastelli, Warped product of Hamiltonians and extensions of Hamiltonian systems. J. Phys.: Conf. Ser. 597, 012024 (2015). https://doi.org/10.1088/1742-6596/597/1/012024
• A. Albouy, There is a projective dynamics. Eur. Math. Soc. Newsl. 89, 37–43 (2013). http://www.ems-ph.org/journals/newsletter/pdf/2013-09-89.pdf
• A. Albouy, Projective dynamics and first integrals. Regul. Chaot. Dyn. 20, 247–276 (2015). https://doi.org/10.1134/S1560354715030041
• A. Ballesteros, F.J. Herranz, F. Musso, The anisotropic oscillator on the 2D sphere and the hyperbolic plane. Nonlinearity 26, 971–990 (2013). http://iopscience.iop.org/article/10.1088/0951-7715/26/4/971/pdf
• A. Ballesteros, A. Blasco, F.J. Herranz, F. Musso, A new integrable anisotropic oscillator on the two-dimensional sphere and the hyperbolic plane. J. Phys. A: Math. Theor. 47, 345204 (2014). http://iopscience.iop.org/article/10.1088/1751-8113/47/34/345204/pdf
• E. Inönü, E.P. Wigner, On the contractions of groups and their representations. Proc. Natl. Acad. Sci. U.S.A. 39, 510–524 (1953). https://doi.org/10.1073/pnas.39.6.510
• F.J. Herranz, M. de Montigny, M.A. del Olmo, M. Santander, Cayley–Klein algebras as graded contractions of so(N + 1). J. Phys. A: Math. Gen. 27, 2515–2526 (1994). https://doi.org/10.1088/0305-4470/27/7/027
• Yaglom, I.M.: A Simple Non-Euclidean Geometry and Its Physical Basis (Springer, New York, 1979)
• N.A. Gromov, V.I. Man’ko, The Jordan–Schwinger representations of Cayley–Klein groups. I. The orthogonal groups. J. Math. Phys. 31, 1047–1053 (1990). https://doi.org/10.1063/1.528781
• B. Doubrovine, S. Novikov, A. Fomenko, Géométrie Contemporaine, Méthodes et Applications First Part (MIR, Moscow, 1982)
• J.M. Jauch, E.L. Hill, On the problem of degeneracy in quantum mechanics. Phys. Rev. 57, 641–645 (1940). https://doi.org/10.1103/PhysRev.57.641
• J.P. Amiet, S. Weigert, Commensurate harmonic oscillators: classical symmetries. J. Math. Phys. 43, 4110–4126 (2002). https://doi.org/10.1063/1.1488672
• M.A. Rodríguez, P. Tempesta, P. Winternitz, Reduction of superintegrable systems: the anisotropic harmonic oscillator. Phys. Rev. E 78, 046608 (2008). https://doi.org/10.1103/PhysRevE.78.046608
• A. Ballesteros, F.J. Herranz, S. Kuru, J. Negro, The anisotropic oscillator on curved spaces: a new exactly solvable model. Ann. Phys. 373, 399–423 (2016). https://doi.org/10.1016/j.aop.2016.07.006
• A. Ballesteros, F.J. Herranz, S. Kuru, J. Negro, Factorization approach to superintegrable systems: formalism and applications. Phys. Atom. Nuclei 80, 389–396 (2017). https://doi.org/10.1134/S1063778817020053
• C.D.J. Fernández, J. Negro, M.A. del Olmo, Group approach to the factorization of the radial oscillator equation. Ann. Phys. 252, 386–412 (1996). https://doi.org/10.1006/aphy.1996.0138
• S. Kuru, J. Negro, Factorizations of one-dimensional classical systems. Ann. Phys. 323, 413–431 (2008). https://doi.org/10.1016/j.aop.2007.10.004
• J.A. Calzada, S. Kuru, J. Negro, M.A. del Olmo, Dynamical algebras of general two-parametric Pöschl–Teller Hamiltonian. Ann. Phys. 327, 808–822 (2012). https://doi.org/10.1016/j.aop.2011.12.014
• E. Celeghini, S. Kuru, J. Negro, M.A. del Olmo, A unified approach to quantum and classical TTW systems based on factorizations. Ann. Phys. 332, 27–37 (2013). https://doi.org/10.1016/j.aop.2013.01.008
• J.A. Calzada, S. Kuru, J. Negro, Superintegrable Lissajous systems on the sphere. Eur. Phys. J. Plus 129, 129–164 (2014). https://doi.org/10.1140/epjp/i2014-14164-5
• N.W. Evans, P.E. Verrier, Superintegrability of the caged anisotropic oscillator. J. Math. Phys. 49, 092902 (2008). https://doi.org/10.1063/1.2988133
• N.W. Evans, Superintegrability in classical mechanics. Phys. Rev. A 41, 5666–5676 (1990). https://doi.org/10.1103/PhysRevA.41.5666
• E.G. Kalnins, G.C. Williams, W. Miller, G.S. Pogosyan, Superintegrability in the three–dimensional Euclidean space. J. Math. Phys. 40, 708–725 (1999). https://doi.org/10.1063/1.532699
• Y.N. Demkov, Symmetry group of the isotropic oscillator. Soviet Phys. JETP 36, 63–66 (1959). http://www.jetp.ac.ru/cgi-bin/dn/e_009_01_0063.pdf
• D.M. Fradkin, Three-dimensional isotropic harmonic oscillator and SU 3. Am. J. Phys. 33, 207–211 (1965). https://doi.org/10.1119/1.1971373
• T.I. Fris, V. Mandrosov, Y.A. Smorodinsky, M. Uhlir, P. Winternitz, On higher symmetries in quantum mechanics. Phys. Lett. 16, 354–356 (1965). https://doi.org/10.1016/0031-9163(65)90885-1
• N.W. Evans, Super-integrability of the Winternitz system. Phys. Lett. A 147, 483–486 (1990). https://doi.org/10.1016/0375-9601(90)90611-Q
• N.W. Evans, Group theory of the Smorodinsky-Winternitz system. J. Math. Phys. 32, 3369–3375 (1991). https://doi.org/10.1063/1.529449
• C. Grosche, G.S. Pogosyan, A.N. Sissakian, Path integral discussion for Smorodinsky–Winternitz potentials I. Two- and three dimensional Euclidean spaces. Fortschr. Phys. 43, 453–521 (1995). https://doi.org/10.1002/prop.2190430602
• K.B. Wolf, C.P. Boyer, The 2:1 anisotropic oscillator, separation of variables and symmetry group in Bargmann space. J. Math. Phys. 16, 2215–2223 (1975). https://doi.org/10.1063/1.522471
• P.W. Higgs, Dynamical symmetries in a spherical geometry I. J. Phys. A: Math. Gen. 12, 309–323 (1979). https://doi.org/10.1088/0305-4470/12/3/006
• M.F. Rañada, M. Santander, On harmonic oscillators on the two-dimensional sphere S 2 and the hyperbolic plane H 2 II. J. Math. Phys. 44, 2149–2167 (2003). https://doi.org/10.1063/1.1560552
• E.G. Kalnins, G.S. Pogosyan, W. Miller, Jr., Completeness of multiseparable superintegrability on the complex 2-sphere. J. Phys. A: Math. Gen. 33, 6791–6806 (2000). https://doi.org/10.1088/0305-4470/33/38/310
• E.G. Kalnins, J.M. Kress, G.S. Pogosyan, W. Miller, Jr., Completeness of superintegrability in two-dimensional constant-curvature spaces. J. Phys. A: Math. Gen. 34, 4705–4720 (2001). https://doi.org/10.1088/0305-4470/34/22/311
• E.G. Kalnins, S. Benenti, W. Miller, Jr., Integrability, Stäckel spaces, and rational potentials. J. Math. Phys. 38, 2345–2365 (1997). https://doi.org/10.1063/1.531977
• P. Saksida, Integrable anharmonic oscillators on spheres and hyperbolic spaces. Nonlinearity 14, 977–994 (2001). https://doi.org/10.1088/0951-7715/14/5/304
• A. Nerssesian, V. Yeghikyan, Anisotropic inharmonic Higgs oscillator and related (MICZ-)Kepler-like systems. J. Phys. A: Math. Theor. 41, 155203 (2008). https://doi.org/10.1088/1751-8113/41/15/155203
• I. Marquette, Generalized MICZ-Kepler system, duality, polynomial, and deformed oscillator algebras. J. Math. Phys. 51, 102105 (2010). https://doi.org/10.1063/1.3496900
• H.I. Leemon, Dynamical symmetries in a spherical geometry II. J. Phys. A: Math. Gen. 12, 489–501 (1979). https://doi.org/10.1088/0305-4470/12/4/009
• Y.M. Hakobyan, G.S. Pogosyan, A.N. Sissakian, S.I. Vinitsky, Isotropic oscillator in a space of constant positive curvature: interbasis expansions. Phys. Atom. Nucl. 62, 623–637 (1999). https://arxiv.org/abs/quant-ph/9710045
• A. Nersessian, G. Pogosyan, Relation of the oscillator and Coulomb systems on spheres and pseudospheres. Phys. Rev. A 63, 020103 (2001). https://doi.org/10.1103/PhysRevA.63.020103
• J.F. Cariñena, M.F. Rañada, M. Santander, M. Senthilvelan, A non-linear oscillator with quasi-harmonic behaviour: two-and n-dimensional oscillators. Nonlinearity 17, 1941–1963 (2004). https://doi.org/10.1088/0951-7715/17/5/019
• A. Ballesteros, A. Enciso, F.J. Herranz, O. Ragnisco, Superintegrability on N-dimensional curved spaces: central potentials, centrifugal terms and monopoles. Ann. Phys. 324, 1219–1233 (2009). https://doi.org/10.1016/j.aop.2009.03.001
• C. Grosche, G.S. Pogosyan, A.N. Sissakian, Path integral discussion for Smorodinsky–Winternitz potentials II. The two- and three-dimensional sphere. Fortschr. Phys. 43, 523–563 (1995). https://doi.org/10.1002/prop.2190430603
• M. Hénon, C. Heiles, The applicability of the third integral of motion: some numerical experiments. Astron. J. 69, 73–79 (1964). https://doi.org/10.1086/109234
• T. Bountis, H. Segur, F. Vivaldi, Integrable Hamiltonian systems and the Painlevé property. Phys. Rev. A 25, 1257–1264 (1982). https://doi.org/10.1103/PhysRevA.25.1257
• Y.F. Chang, M. Tabor, J. Weiss, Analytic structure of the Hénon–Heiles Hamiltonian in integrable and nonintegrable regimes. J. Math. Phys. 23, 531–538 (1982). https://doi.org/10.1063/1.525389
• B. Grammaticos, B. Dorizzi, R. Padjen, Painlevé property and integrals of motion for the Hénon–Heiles system. Phys. Lett. A 89, 111–113 (1982). https://doi.org/10.1016/0375-9601(82)90868-4
• J. Hietarinta, Integrable families of Hénon–Heiles-type Hamiltonians and a new duality. Phys. Rev. A 28, 3670–3672 (1983). https://doi.org/10.1103/PhysRevA.28.3670
• A.P. Fordy, Hamiltonian symmetries of the Hénon–Heiles system. Phys. Lett. A 97, 21–23 (1983). https://doi.org/10.1016/0375-9601(83)90091-9
• S. Wojciechowski, Separability of an integrable case of the Hénon–Heiles system. Phys. Lett. A 100, 277–278 (1984). https://doi.org/10.1016/0375-9601(84)90535-8
• R. Sahadevan, M. Lakshmanan, Invariance and integrability: Hénon–Heiles and two coupled quartic anharmonic oscillator systems. J. Phys. A: Math. Gen. 19, L949–L954 (1986). https://doi.org/10.1088/0305-4470/19/16/001
• A.P. Fordy, The Hénon–Heiles system revisited. Phys. D 52, 204–210 (1991). https://doi.org/10.1016/0167-2789(91)90122-P
• W. Sarlet, New aspects of integrability of generalized Hénon–Heiles systems. J. Phys. A: Math. Gen. 24, 5245–5251 (1991). https://doi.org/10.1088/0305-4470/24/22/008
• V. Ravoson, L. Gavrilov, R. Caboz, Separability and Lax pairs for Hénon–Heiles system. J. Math. Phys. 34, 2385–2393 (1993). https://doi.org/10.1063/1.530123
• G. Tondo, On the integrability of stationary and restricted flows of the KdV hierarchy. J. Phys. A: Math. Gen. 28, 5097–5115 (1995). https://doi.org/10.1088/0305-4470/28/17/034
• R. Conte, M. Musette, C. Verhoeven, Completeness of the cubic and quartic Hénon–Heiles Hamiltonians. Theor. Math. Phys. 144, 888–898 (2005). https://doi.org/10.1007/s11232-005-0115-9
• A. Ballesteros, A. Blasco, F.J. Herranz, A curved Hénon-Heiles system and its integrable perturbations. J. Phys.: Conf. Ser. 597, 012013 (2015). https://doi.org/10.1088/1742-6596/597/1/012013
• A. Ramani, B. Dorizzi, B. Grammaticos, Painlevé conjecture revisited. Phys. Rev. Lett. 49, 1539–1541 (1982). https://doi.org/10.1103/PhysRevLett.49.1539
• J. Hietarinta, Direct method for the search of the second invariant. Phys. Rep. 147, 87–154 (1987). https://doi.org/10.1016/0370-1573(87)90089-5
• E.V. Ferapontov, A.P. Fordy, Separable Hamiltonians and integrable systems of hydrodynamic type. J. Geom. Phys. 21, 169–182 (1997). https://doi.org/10.1016/S0393-0440(96)00013-7
• A.N.W. Hone, V. Novikov, C. Verhoeven, An integrable hierarchy with a perturbed Hénon–Heiles system. Inv. Probl. 22, 2001–2020 (2006). https://doi.org/10.1088/0266-5611/22/6/006
• A.N.W. Hone, V. Novikov, C. Verhoeven, An extended Hénon–Heiles system. Phys. Lett. A 372, 1440–1444 (2008). https://doi.org/10.1016/j.physleta.2007.09.063
• A. Blasco, Integrability of non-linear Hamiltonian systems with N degrees of freedom. Ph.D. Thesis, Burgos University, Burgos, 2009. http://riubu.ubu.es/bitstream/10259/106/4/Blasco_Sanz.pdf
• A. Ballesteros, A. Blasco, Integrable Hénon–Heiles Hamiltonians: a Poisson algebra approach. Ann. Phys. 325, 2787–2799 (2010). https://doi.org/10.1016/j.aop.2010.08.002
• A. Ballesteros, A. Blasco, F.J. Herranz, F. Musso, An integrable Hénon–Heiles system on the sphere and the hyperbolic plane. Nonlinearity 28, 3789–3801 (2015). https://doi.org/10.1088/0951-7715/28/11/3789
• P. Serret, Théorie nouvelle géométrique et mécanique des lignes à double courbure. Paris, Mallet-Bachelier (1859)
• D.R. Petrosyan, G.S. Pogosyan, Harmonic oscillator on the SO(2,2) hyperboloid. SIGMA Symmetry Integrability Geom. Methods Appl. 11, 096 (2015). https://doi.org/10.3842/SIGMA.2015.096
• E.G. Kalnins, J.M. Kress, P. Winternitz, Superintegrability in a two-dimensional space of nonconstant curvature. J. Math. Phys. 43, 970–983 (2002). https://doi.org/10.1063/1.1429322
• A. Ballesteros, F.J. Herranz, O. Ragnisco, Integrable potentials on spaces with curvature from quantum groups. J. Phys. A: Math. Theor. 38, 7129–7144 (2005). https://doi.org/10.1088/0305-4470/38/32/004
• A. Ballesteros, A. Enciso, F.J. Herranz, O. Ragnisco, A maximally superintegrable system on an n-dimensional space of nonconstant curvature. Phys. D 237, 505–509 (2008). https://doi.org/10.1016/j.physd.2007.09.021
• A. Ballesteros, A. Enciso, F.J. Herranz, O. Ragnisco, Bertrand spacetimes as Kepler/oscillator potentials. Class. Quant. Grav. 25, 165005 (2008). https://doi.org/10.1088/0264-9381/25/16/165005
• A. Ballesteros, A. Enciso, F.J. Herranz, O. Ragnisco, Hamiltonian systems admitting a Runge-Lenz vector and an optimal extension of Bertrand’s theorem to curved manifolds. Commun. Math. Phys. 290, 1033–1049 (2009). https://doi.org/10.1007/s00220-009-0793-5
• A. Ballesteros, A. Blasco, F.J. Herranz, F. Musso, O. Ragnisco, (Super)integrability from coalgebra symmetry: formalism and applications. J. Phys.: Conf. Ser. 175, 012004 (2009). https://doi.org/10.1088/1742-6596/175/1/012004
• O. Ragnisco, D. Riglioni, A family of exactly solvable radial quantum systems on space of non-constant curvature with accidental degeneracy in the spectrum. SIGMA Symmetry Integrability Geom. Methods Appl. 6, 097 (2010). https://doi.org/10.3842/SIGMA.2010.097
• A. Ballesteros, A. Enciso, F.J. Herranz, O. Ragnisco, O. Riglioni, Superintegrable oscillator and Kepler systems on spaces of nonconstant curvature via the Säckel Transform. SIGMA Symmetry Integrability Geom. Methods Appl. 7, 048 (2011). https://doi.org/10.3842/SIGMA.2011.048
• A. Ballesteros, A. Enciso, F.J. Herranz, O. Ragnisco, O. Riglioni, Quantum mechanics on spaces of nonconstant curvature: the oscillator problem and superintegrability. Ann. Phys. 326, 2053–2073 (2011). https://doi.org/10.1016/j.aop.2011.03.002
• A. Ballesteros, A. Enciso, F.J. Herranz, O. Ragnisco, D. Riglioni, An exactly solvable deformation of the Coulomb problem associated with the Taub-NUT metric. Ann. Phys. 351, 540–577 (2014). https://doi.org/10.1016/j.aop.2014.09.013