June  2021, 13(2): 195-208. 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  June 2021 Early access  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, 2021, 13 (2) : 195-208. doi: 10.3934/jgm.2021003
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.

[2]

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

[3]

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

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[15]

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

[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.

[17]

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

[18]

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

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[27]

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

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[41]

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

[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.

[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.

[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.

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.

[2]

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

[3]

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

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[15]

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

[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.

[17]

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

[18]

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

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[27]

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

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[41]

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

[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.

[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.

[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.

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