doi: 10.3934/jgm.2021003

Quasi-bi-Hamiltonian structures and superintegrability: Study of a Kepler-related family of systems endowed with generalized Runge-Lenz integrals of motion

Departamento de Física Teórica and IUMA, Universidad de Zaragoza, 50009 Zaragoza, Spain

Received  October 2020 Revised  January 2021 Published  February 2021

The existence of quasi-bi-Hamiltonian structures for a two-dimen-sional superintegrable $ (k_1,k_2,k_3) $-dependent Kepler-related problem is studied. We make use of an approach that is related with the existence of some complex functions which satisfy interesting Poisson bracket relations and that was previously applied to the standard Kepler problem as well as to some particular superintegrable systems as the Smorodinsky-Winternitz (SW) system, the Tremblay-Turbiner-Winternitz (TTW) and Post-Winternitz (PW) systems. We prove that these complex functions are important for two reasons: first, they determine the integrals of motion, and second they determine the existence of some geometric structures (in this particular case, quasi-bi-Hamiltonian structures). All the results depend on three parameters ($ k_1, k_2, k_3 $) in such a way that in the particular case $ k_1\ne 0 $, $ k_2 = k_3 = 0 $, the properties characterizing the Kepler problem are obtained.

This paper can be considered as divided in two parts and every part presents a different approach (different complex functions and different quasi-bi-Hamil-tonian structures).

Citation: Manuel F. Rañada. Quasi-bi-Hamiltonian structures and superintegrability: Study of a Kepler-related family of systems endowed with generalized Runge-Lenz integrals of motion. Journal of Geometric Mechanics, doi: 10.3934/jgm.2021003
References:
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Á. BallesterosA. EncisoF. J. Herranz and O. Ragnisco, Hamiltonian systems admitting a Runge-Lenz vector and an optimal extension of Bertrand's theorem to curved manifolds, Comm. Math. Phys., 290 (2009), 1033-1049.  doi: 10.1007/s00220-009-0793-5.  Google Scholar

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U. Ben-Ya'acov, Laplace-Runge-Lenz symmetry in general rotationally symmetric systems, J. Math. Phys., 51 (2010). doi: 10.1063/1.3520521.  Google Scholar

[3]

M. Blaszak, Bi-Hamiltonian representation of Stäckel systems, Phys. Rev. E (3), 79 (2009). doi: 10.1103/PhysRevE.79.056607.  Google Scholar

[4]

H. BoualemR. Brouzet and J. Rakotondralambo, About the separability of completely integrable quasi-bi-Hamiltonian systems with compact levels, Differential Geom. Appl., 26 (2008), 583-591.  doi: 10.1016/j.difgeo.2008.04.008.  Google Scholar

[5]

H. BoualemR. Brouzet and J. Rakotondralambo, Quasi-bi-Hamiltonian systems: Why the Pfaffian case?, Phys. Lett. A, 359 (2006), 559-563.  doi: 10.1016/j.physleta.2006.07.019.  Google Scholar

[6]

R. BrouzetR. CabozJ. Rabenivo and V. Ravoson, Two degrees of freedom quasi bi-Hamiltonian systems, J. Phys. A, 29 (1996), 2069-2076.  doi: 10.1088/0305-4470/29/9/019.  Google Scholar

[7]

J. F. Cariñena, P. Guha and M. F. Rañada, Hamiltonian and quasi-Hamiltonian systems, Nambu-Poisson structures and symmetries, J. Phys. A, 41 (2008), 11pp. doi: 10.1088/1751-8113/41/33/335209.  Google Scholar

[8]

J. F. CariñenaP. Guha and M. F. Rañada, Quasi-Hamiltonian structure and Hojman construction, J. Math. Anal. Appl., 332 (2007), 975-988.  doi: 10.1016/j.jmaa.2006.08.092.  Google Scholar

[9]

J. F. Cariñena, G. Marmo and M. F. Rañada, Non-symplectic symmetries and bi-Hamiltonian structures of the rational harmonic oscillator, J. Phys. A, 35 (2002), L679–L686. doi: 10.1088/0305-4470/35/47/101.  Google Scholar

[10]

J. F. Cariñena and M. F. Rañada, Canonoid transformations from a geometric perspective, J. Math. Phys., 29 (1988), 2181-2186.  doi: 10.1063/1.528146.  Google Scholar

[11]

J. F. Cariñena and M. F. Rañada, Quasi-bi-Hamiltonian structures of the 2-dimensional Kepler problem, SIGMA Symmetry Integrability Geom. Methods Appl., 12 (2016), 16pp. doi: 10.3842/SIGMA.2016.010.  Google Scholar

[12]

J. F. CariñenaM. F. Rañada and M. Santander, The Kepler problem and the Laplace-Runge-Lenz vector on spaces of constant curvature and arbitrary signature, Qual. Theory Dyn. Syst., 7 (2008), 87-99.  doi: 10.1007/s12346-008-0004-3.  Google Scholar

[13]

P. Casati, F. Magri and M. Pedroni, The bi-Hamiltonian approach to integrable systems, in Modern Group Analysis: Advanced Analytical and Computational Methods in Mathematical Physics, Kluwer Acad. Publ., Dordrecht, 1993,101–110.  Google Scholar

[14]

C. M. Chanu and G. Rastelli, On the extended-Hamiltonian structure of certain superintegrable systems on constant-curvature Riemannian and pseudo-Riemannian surfaces, SIGMA Symmetry Integrability Geom. Methods Appl., 16 (2020), 16pp. doi: 10.3842/SIGMA.2020.052.  Google Scholar

[15]

M. Crampin and W. Sarlet, Bi-quasi-Hamiltonian systems, J. Math. Phys., 43 (2002), 2505-2517.  doi: 10.1063/1.1462856.  Google Scholar

[16]

M. Crampin, W. Sarlet and G. Thompson, Bi-differential calculi and bi-Hamiltonian systems, J. Phys. A, 33 (2000), L177–L180. doi: 10.1088/0305-4470/33/20/101.  Google Scholar

[17]

N. W. Evans, Superintegrability in classical mechanics, Phys. Rev. A (3), 41 (1990), 5666-5676.  doi: 10.1103/PhysRevA.41.5666.  Google Scholar

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R. L. Fernandes, Completely integrable bi-Hamiltonian systems, J. Dynam. Differential Equations, 6 (1994), 53-69.  doi: 10.1007/BF02219188.  Google Scholar

[19]

A. P. Fordy and Q. Huang, Superintegrable systems on 3 dimensional conformally flat spaces, J. Geom. Phys., 153 (2020), 27pp. doi: 10.1016/j.geomphys.2020.103687.  Google Scholar

[20]

T. I. FrišV. MandrosovY. A. SmorodinskyM. Uhliř and P. Winternitz, On higher symmetries in quantum mechanics, Phys. Lett., 16 (1965), 354-356.  doi: 10.1016/0031-9163(65)90885-1.  Google Scholar

[21]

C. Gonera and J. Gonera, New superintegrable models on spaces of constant curvature, Ann. Physics, 413 (2020), 16pp. doi: 10.1016/j.aop.2019.168052.  Google Scholar

[22]

Y. A. Grigoriev and A. V. Tsiganov, On superintegrable systems separable in Cartesian coordinates, Phys. Lett. A, 382 (2018), 2092-2096.  doi: 10.1016/j.physleta.2018.05.039.  Google Scholar

[23]

C. GroscheG. S. Pogosyan and A. N. Sissakian, Path integral discussion for Smorodinsky-Winternitz potentials. I. Two- and three-dimensional Euclidean spaces, Fortschr. Phys., 43 (1995), 453-521.  doi: 10.1002/prop.2190430602.  Google Scholar

[24]

A. Holas and N. H. March, A generalisation of the Runge-Lenz constant of classical motion in a central potential, J. Phys. A, 23 (1990), 735-749.  doi: 10.1088/0305-4470/23/5/017.  Google Scholar

[25]

J. M. Jauch and E. L. Hill, On the problem of degeneracy in quantum mechanics, Phys. Rev., 57 (1940), 641-645.  doi: 10.1103/PhysRev.57.641.  Google Scholar

[26]

P. G. L. Leach and G. P. Flessas, Generalisations of the Laplace-Runge-Lenz vector, J. Nonlinear Math. Phys., 10 (2003), 340-423.  doi: 10.2991/jnmp.2003.10.3.6.  Google Scholar

[27]

I. Marquette, Generalized MICZ-Kepler system, duality, polynomial, and deformed oscillator algebras, J. Math. Phys., 51 (2010), 10pp. doi: 10.1063/1.3496900.  Google Scholar

[28]

W. Miller Jr., S. Post and P. Winternitz, Classical and quantum superintegrability with applications, J. Phys. A, 46 (2013), 97pp. doi: 10.1088/1751-8113/46/42/423001.  Google Scholar

[29]

C. Morosi and G. Tondo, On a class of dynamical systems both quasi-bi-Hamiltonian and bi-Hamiltonian, Phys. Lett. A, 247 (1998), 59-64.  doi: 10.1016/S0375-9601(98)00543-X.  Google Scholar

[30]

C. Morosi and G. Tondo, Quasi-bi-Hamiltonian systems and separability, J. Phys. A, 30 (1997), 2799-2806.  doi: 10.1088/0305-4470/30/8/023.  Google Scholar

[31]

A. G. Nikitin, Laplace-Runge-Lenz vector with spin in any dimension, J. Phys. A, 47 (2014), 16pp. doi: 10.1088/1751-8113/47/37/375201.  Google Scholar

[32]

A. M. Perelomov, Integrable Systems of Classical Mechanics and Lie Algebras. Vol. I, Birkhäuser Verlag, Basel, 1990. doi: 10.1007/978-3-0348-9257-5.  Google Scholar

[33]

S. Post and P. Winternitz, An infinite family of superintegrable deformations of the Coulomb potential, J. Phys. A, 43 (2010), 11pp. doi: 10.1088/1751-8113/43/22/222001.  Google Scholar

[34]

M. F. Rañada, Bi-Hamiltonian structure of the bi-dimensional superintegrable nonlinear isotonic oscillator, J. Math. Phys., 57 (2016), 13pp. doi: 10.1063/1.4948641.  Google Scholar

[35]

M. F. Rañada, Dynamical symmetries, bi-Hamiltonian structures, and superintegrable $n = 2$ systems, J. Math. Phys., 41 (2000), 2121-2134.  doi: 10.1063/1.533230.  Google Scholar

[36]

M. F. Rañada, 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: 10.1016/j.physleta.2015.07.043.  Google Scholar

[37]

M. F. Rañada, Quasi-bi-Hamiltonian structures, complex functions and superintegrability: The Tremblay-Turbiner-Winternitz (TTW) and the Post-Winternitz (PW) systems, J. Phys. A, 50 (2017), 19pp. doi: 10.1088/1751-8121/aa7951.  Google Scholar

[38]

M. F. Rañada, Superintegrable $n = 2$ systems, quadratic constants of motion, and potentials of Drach, J. Math. Phys., 38 (1997), 4165-4178.  doi: 10.1063/1.532089.  Google Scholar

[39]

M. F. Rañada, The Tremblay-Turbiner-Winternitz system on spherical and hyperbolic spaces: Superintegrability, curvature-dependent formalism and complex factorization, J. Phys. A, 47 (2014), 9pp. doi: 10.1088/1751-8113/47/16/165203.  Google Scholar

[40]

M. F. Rañada, M. A. Rodriguez and M. Santander, A new proof of the higher-order superintegrability of a noncentral oscillator with inversely quadratic nonlinearities, J. Math. Phys., 51 (2010), 11pp. doi: 10.1063/1.3374665.  Google Scholar

[41]

V. Ravoson, Structure Bihamiltonienne, Séparabilité, Paires de Lax et Integrabilité, Ph.D thesis, Univ. de Pau, 1992. Google Scholar

[42]

M. A. Rodriguez, P. Tempesta and P. Winternitz, Symmetry reduction and superintegrable Hamiltonian systems, J. Phys. Conf. Ser., 175 (2009). doi: 10.1088/1742-6596/175/1/012013.  Google Scholar

[43]

H. White, On a class of dynamical systems admitting both Poincaré and Laplace-Runge-Lenz vectors, Nuovo Cimento Soc. Ital. Fis. B, 125 (2010), 7-25.  doi: 10.1393/ncb/i2010-10837-y.  Google Scholar

[44]

Y. B. Zeng and W.-X. Ma, Families of quasi-bi-Hamiltonian systems and separability, J. Math. Phys., 40 (1999), 4452-4473.  doi: 10.1063/1.532979.  Google Scholar

show all references

References:
[1]

Á. BallesterosA. EncisoF. J. Herranz and O. Ragnisco, Hamiltonian systems admitting a Runge-Lenz vector and an optimal extension of Bertrand's theorem to curved manifolds, Comm. Math. Phys., 290 (2009), 1033-1049.  doi: 10.1007/s00220-009-0793-5.  Google Scholar

[2]

U. Ben-Ya'acov, Laplace-Runge-Lenz symmetry in general rotationally symmetric systems, J. Math. Phys., 51 (2010). doi: 10.1063/1.3520521.  Google Scholar

[3]

M. Blaszak, Bi-Hamiltonian representation of Stäckel systems, Phys. Rev. E (3), 79 (2009). doi: 10.1103/PhysRevE.79.056607.  Google Scholar

[4]

H. BoualemR. Brouzet and J. Rakotondralambo, About the separability of completely integrable quasi-bi-Hamiltonian systems with compact levels, Differential Geom. Appl., 26 (2008), 583-591.  doi: 10.1016/j.difgeo.2008.04.008.  Google Scholar

[5]

H. BoualemR. Brouzet and J. Rakotondralambo, Quasi-bi-Hamiltonian systems: Why the Pfaffian case?, Phys. Lett. A, 359 (2006), 559-563.  doi: 10.1016/j.physleta.2006.07.019.  Google Scholar

[6]

R. BrouzetR. CabozJ. Rabenivo and V. Ravoson, Two degrees of freedom quasi bi-Hamiltonian systems, J. Phys. A, 29 (1996), 2069-2076.  doi: 10.1088/0305-4470/29/9/019.  Google Scholar

[7]

J. F. Cariñena, P. Guha and M. F. Rañada, Hamiltonian and quasi-Hamiltonian systems, Nambu-Poisson structures and symmetries, J. Phys. A, 41 (2008), 11pp. doi: 10.1088/1751-8113/41/33/335209.  Google Scholar

[8]

J. F. CariñenaP. Guha and M. F. Rañada, Quasi-Hamiltonian structure and Hojman construction, J. Math. Anal. Appl., 332 (2007), 975-988.  doi: 10.1016/j.jmaa.2006.08.092.  Google Scholar

[9]

J. F. Cariñena, G. Marmo and M. F. Rañada, Non-symplectic symmetries and bi-Hamiltonian structures of the rational harmonic oscillator, J. Phys. A, 35 (2002), L679–L686. doi: 10.1088/0305-4470/35/47/101.  Google Scholar

[10]

J. F. Cariñena and M. F. Rañada, Canonoid transformations from a geometric perspective, J. Math. Phys., 29 (1988), 2181-2186.  doi: 10.1063/1.528146.  Google Scholar

[11]

J. F. Cariñena and M. F. Rañada, Quasi-bi-Hamiltonian structures of the 2-dimensional Kepler problem, SIGMA Symmetry Integrability Geom. Methods Appl., 12 (2016), 16pp. doi: 10.3842/SIGMA.2016.010.  Google Scholar

[12]

J. F. CariñenaM. F. Rañada and M. Santander, The Kepler problem and the Laplace-Runge-Lenz vector on spaces of constant curvature and arbitrary signature, Qual. Theory Dyn. Syst., 7 (2008), 87-99.  doi: 10.1007/s12346-008-0004-3.  Google Scholar

[13]

P. Casati, F. Magri and M. Pedroni, The bi-Hamiltonian approach to integrable systems, in Modern Group Analysis: Advanced Analytical and Computational Methods in Mathematical Physics, Kluwer Acad. Publ., Dordrecht, 1993,101–110.  Google Scholar

[14]

C. M. Chanu and G. Rastelli, On the extended-Hamiltonian structure of certain superintegrable systems on constant-curvature Riemannian and pseudo-Riemannian surfaces, SIGMA Symmetry Integrability Geom. Methods Appl., 16 (2020), 16pp. doi: 10.3842/SIGMA.2020.052.  Google Scholar

[15]

M. Crampin and W. Sarlet, Bi-quasi-Hamiltonian systems, J. Math. Phys., 43 (2002), 2505-2517.  doi: 10.1063/1.1462856.  Google Scholar

[16]

M. Crampin, W. Sarlet and G. Thompson, Bi-differential calculi and bi-Hamiltonian systems, J. Phys. A, 33 (2000), L177–L180. doi: 10.1088/0305-4470/33/20/101.  Google Scholar

[17]

N. W. Evans, Superintegrability in classical mechanics, Phys. Rev. A (3), 41 (1990), 5666-5676.  doi: 10.1103/PhysRevA.41.5666.  Google Scholar

[18]

R. L. Fernandes, Completely integrable bi-Hamiltonian systems, J. Dynam. Differential Equations, 6 (1994), 53-69.  doi: 10.1007/BF02219188.  Google Scholar

[19]

A. P. Fordy and Q. Huang, Superintegrable systems on 3 dimensional conformally flat spaces, J. Geom. Phys., 153 (2020), 27pp. doi: 10.1016/j.geomphys.2020.103687.  Google Scholar

[20]

T. I. FrišV. MandrosovY. A. SmorodinskyM. Uhliř and P. Winternitz, On higher symmetries in quantum mechanics, Phys. Lett., 16 (1965), 354-356.  doi: 10.1016/0031-9163(65)90885-1.  Google Scholar

[21]

C. Gonera and J. Gonera, New superintegrable models on spaces of constant curvature, Ann. Physics, 413 (2020), 16pp. doi: 10.1016/j.aop.2019.168052.  Google Scholar

[22]

Y. A. Grigoriev and A. V. Tsiganov, On superintegrable systems separable in Cartesian coordinates, Phys. Lett. A, 382 (2018), 2092-2096.  doi: 10.1016/j.physleta.2018.05.039.  Google Scholar

[23]

C. GroscheG. S. Pogosyan and A. N. Sissakian, Path integral discussion for Smorodinsky-Winternitz potentials. I. Two- and three-dimensional Euclidean spaces, Fortschr. Phys., 43 (1995), 453-521.  doi: 10.1002/prop.2190430602.  Google Scholar

[24]

A. Holas and N. H. March, A generalisation of the Runge-Lenz constant of classical motion in a central potential, J. Phys. A, 23 (1990), 735-749.  doi: 10.1088/0305-4470/23/5/017.  Google Scholar

[25]

J. M. Jauch and E. L. Hill, On the problem of degeneracy in quantum mechanics, Phys. Rev., 57 (1940), 641-645.  doi: 10.1103/PhysRev.57.641.  Google Scholar

[26]

P. G. L. Leach and G. P. Flessas, Generalisations of the Laplace-Runge-Lenz vector, J. Nonlinear Math. Phys., 10 (2003), 340-423.  doi: 10.2991/jnmp.2003.10.3.6.  Google Scholar

[27]

I. Marquette, Generalized MICZ-Kepler system, duality, polynomial, and deformed oscillator algebras, J. Math. Phys., 51 (2010), 10pp. doi: 10.1063/1.3496900.  Google Scholar

[28]

W. Miller Jr., S. Post and P. Winternitz, Classical and quantum superintegrability with applications, J. Phys. A, 46 (2013), 97pp. doi: 10.1088/1751-8113/46/42/423001.  Google Scholar

[29]

C. Morosi and G. Tondo, On a class of dynamical systems both quasi-bi-Hamiltonian and bi-Hamiltonian, Phys. Lett. A, 247 (1998), 59-64.  doi: 10.1016/S0375-9601(98)00543-X.  Google Scholar

[30]

C. Morosi and G. Tondo, Quasi-bi-Hamiltonian systems and separability, J. Phys. A, 30 (1997), 2799-2806.  doi: 10.1088/0305-4470/30/8/023.  Google Scholar

[31]

A. G. Nikitin, Laplace-Runge-Lenz vector with spin in any dimension, J. Phys. A, 47 (2014), 16pp. doi: 10.1088/1751-8113/47/37/375201.  Google Scholar

[32]

A. M. Perelomov, Integrable Systems of Classical Mechanics and Lie Algebras. Vol. I, Birkhäuser Verlag, Basel, 1990. doi: 10.1007/978-3-0348-9257-5.  Google Scholar

[33]

S. Post and P. Winternitz, An infinite family of superintegrable deformations of the Coulomb potential, J. Phys. A, 43 (2010), 11pp. doi: 10.1088/1751-8113/43/22/222001.  Google Scholar

[34]

M. F. Rañada, Bi-Hamiltonian structure of the bi-dimensional superintegrable nonlinear isotonic oscillator, J. Math. Phys., 57 (2016), 13pp. doi: 10.1063/1.4948641.  Google Scholar

[35]

M. F. Rañada, Dynamical symmetries, bi-Hamiltonian structures, and superintegrable $n = 2$ systems, J. Math. Phys., 41 (2000), 2121-2134.  doi: 10.1063/1.533230.  Google Scholar

[36]

M. F. Rañada, 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: 10.1016/j.physleta.2015.07.043.  Google Scholar

[37]

M. F. Rañada, Quasi-bi-Hamiltonian structures, complex functions and superintegrability: The Tremblay-Turbiner-Winternitz (TTW) and the Post-Winternitz (PW) systems, J. Phys. A, 50 (2017), 19pp. doi: 10.1088/1751-8121/aa7951.  Google Scholar

[38]

M. F. Rañada, Superintegrable $n = 2$ systems, quadratic constants of motion, and potentials of Drach, J. Math. Phys., 38 (1997), 4165-4178.  doi: 10.1063/1.532089.  Google Scholar

[39]

M. F. Rañada, The Tremblay-Turbiner-Winternitz system on spherical and hyperbolic spaces: Superintegrability, curvature-dependent formalism and complex factorization, J. Phys. A, 47 (2014), 9pp. doi: 10.1088/1751-8113/47/16/165203.  Google Scholar

[40]

M. F. Rañada, M. A. Rodriguez and M. Santander, A new proof of the higher-order superintegrability of a noncentral oscillator with inversely quadratic nonlinearities, J. Math. Phys., 51 (2010), 11pp. doi: 10.1063/1.3374665.  Google Scholar

[41]

V. Ravoson, Structure Bihamiltonienne, Séparabilité, Paires de Lax et Integrabilité, Ph.D thesis, Univ. de Pau, 1992. Google Scholar

[42]

M. A. Rodriguez, P. Tempesta and P. Winternitz, Symmetry reduction and superintegrable Hamiltonian systems, J. Phys. Conf. Ser., 175 (2009). doi: 10.1088/1742-6596/175/1/012013.  Google Scholar

[43]

H. White, On a class of dynamical systems admitting both Poincaré and Laplace-Runge-Lenz vectors, Nuovo Cimento Soc. Ital. Fis. B, 125 (2010), 7-25.  doi: 10.1393/ncb/i2010-10837-y.  Google Scholar

[44]

Y. B. Zeng and W.-X. Ma, Families of quasi-bi-Hamiltonian systems and separability, J. Math. Phys., 40 (1999), 4452-4473.  doi: 10.1063/1.532979.  Google Scholar

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