Geodesic
On the Extended-Hamiltonian Structure of Certain Superintegrable Systems on Constant-Curvature Riemannian and Pseudo-Riemannian Surfaces
Symmetry, integrability and geometry: methods and applications, Tome 16 (2020) Cet article a éte moissonné depuis la source Math-Net.Ru

Voir la notice de l'article

We prove the integrability and superintegrability of a family of natural Hamiltonians which includes and generalises those studied in some literature, originally defined on the 2D Minkowski space. Some of the new Hamiltonians are a perfect analogy of the well-known superintegrable system on the Euclidean plane proposed by Tremblay–Turbiner–Winternitz and they are defined on Minkowski space, as well as on all other 2D manifolds of constant curvature, Riemannian or pseudo-Riemannian. We show also how the application of the coupling-constant-metamorphosis technique allows us to obtain new superintegrable Hamiltonians from the previous ones. Moreover, for the Minkowski case, we show the quantum superintegrability of the corresponding quantum Hamiltonian operator. Our results are obtained by applying the theory of extended Hamiltonian systems, which is strictly connected with the geometry of warped manifolds.
Keywords: extended-Hamiltonian, superintegrable systems, constant curvature. 
@article{SIGMA_2020_16_a51,
     author = {Claudia Maria Chanu and Giovanni Rastelli},
     title = {On the {Extended-Hamiltonian} {Structure} of {Certain} {Superintegrable} {Systems} on {Constant-Curvature} {Riemannian} and {Pseudo-Riemannian} {Surfaces}},
     journal = {Symmetry, integrability and geometry: methods and applications},
     year = {2020},
     volume = {16},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/SIGMA_2020_16_a51/}
}
TY  - JOUR
AU  - Claudia Maria Chanu
AU  - Giovanni Rastelli
TI  - On the Extended-Hamiltonian Structure of Certain Superintegrable Systems on Constant-Curvature Riemannian and Pseudo-Riemannian Surfaces
JO  - Symmetry, integrability and geometry: methods and applications
PY  - 2020
VL  - 16
UR  - http://geodesic.mathdoc.fr/item/SIGMA_2020_16_a51/
LA  - en
ID  - SIGMA_2020_16_a51
ER  - 
%0 Journal Article
%A Claudia Maria Chanu
%A Giovanni Rastelli
%T On the Extended-Hamiltonian Structure of Certain Superintegrable Systems on Constant-Curvature Riemannian and Pseudo-Riemannian Surfaces
%J Symmetry, integrability and geometry: methods and applications
%D 2020
%V 16
%U http://geodesic.mathdoc.fr/item/SIGMA_2020_16_a51/
%G en
%F SIGMA_2020_16_a51
Claudia Maria Chanu; Giovanni Rastelli. On the Extended-Hamiltonian Structure of Certain Superintegrable Systems on Constant-Curvature Riemannian and Pseudo-Riemannian Surfaces. Symmetry, integrability and geometry: methods and applications, Tome 16 (2020). http://geodesic.mathdoc.fr/item/SIGMA_2020_16_a51/

[1] Benenti S., “Intrinsic characterization of the variable separation in the Hamilton–Jacobi equation”, J. Math. Phys., 38 (1997), 6578–6602 | DOI | MR | Zbl

[2] Campoamor-Stursberg R., “Superposition of super-integrable pseudo-Euclidean potentials in $N=2$ with a fundamental constant of motion of arbitrary order in the momenta”, J. Math. Phys., 55 (2014), 042904, 11 pp. | DOI | MR | Zbl

[3] Chanu C. M., Degiovanni L., Rastelli G., “First integrals of extended Hamiltonians in $n+1$ dimensions generated by powers of an operator”, SIGMA, 7 (2011), 038, 12 pp., arXiv: 1101.5975 | DOI | MR | Zbl

[4] Chanu C. M., Degiovanni L., Rastelli G., “Polynomial constants of motion for Calogero-type systems in three dimensions”, J. Math. Phys., 52 (2011), 032903, 7 pp., arXiv: 1002.2735 | DOI | MR | Zbl

[5] Chanu C. M., Degiovanni L., Rastelli G., “Generalizations of a method for constructing first integrals of a class of natural Hamiltonians and some remarks about quantization”, J. Phys. Conf. Ser., 343 (2012), 012101, 15 pp., arXiv: 1111.0030 | DOI

[6] Chanu C. M., Degiovanni L., Rastelli G., “Superintegrable extensions of superintegrable systems”, SIGMA, 8 (2012), 070, 12 pp., arXiv: 1210.3126 | DOI | MR | Zbl

[7] Chanu C. M., Degiovanni L., Rastelli G., “Extensions of Hamiltonian systems dependent on a rational parameter”, J. Math. Phys., 55 (2014), 122703, 11 pp., arXiv: 1310.5690 | DOI | MR | Zbl

[8] Chanu C. M., Degiovanni L., Rastelli G., “The Tremblay–Turbiner–Winternitz system as extended Hamiltonian”, J. Math. Phys., 55 (2014), 122701, 8 pp., arXiv: 1404.4825 | DOI | MR | Zbl

[9] Chanu C. M., Degiovanni L., Rastelli G., “Extended Hamiltonians, coupling-constant metamorphosis and the Post–Winternitz system”, SIGMA, 11 (2015), 094, 9 pp., arXiv: 1509.07288 | DOI | MR | Zbl

[10] Chanu C. M., Rastelli G., “Extended Hamiltonians and shift, ladder functions and operators”, Ann. Physics, 386 (2017), 254–274, arXiv: 1705.09519 | DOI | MR | Zbl

[11] Chanu C. M., Rastelli G., “Extensions of non-natural Hamiltonians”, Theoret. and Math. Phys. (to appear) , arXiv: 2001.04152 | MR

[12] Hietarinta J., Grammaticos B., Dorizzi B., Ramani A., “Coupling-constant metamorphosis and duality between integrable Hamiltonian systems”, Phys. Rev. Lett., 53 (1984), 1707–1710 | DOI | MR

[13] Kalnins E. G., Kress J. M., Miller Jr. W., Separation of variables and superintegrability. The symmetry of solvable systems, IOP Expanding Physics, IOP Publishing, Bristol, 2018 | DOI | MR | Zbl

[14] Kalnins E. G., Miller Jr. W., Post S., “Coupling constant metamorphosis and $N$th-order symmetries in classical and quantum mechanics”, J. Phys. A: Math. Theor., 43 (2010), 035202, 20 pp., arXiv: 0908.4393 | DOI | MR | Zbl

[15] Kuru Ş, Negro J., “Factorizations of one-dimensional classical systems”, Ann. Physics, 323 (2008), 413–431, arXiv: 0709.4649 | DOI | MR | Zbl

[16] Maciejewski A. J., Przybylska M., Tsiganov A. V., “On algebraic construction of certain integrable and super-integrable systems”, Phys. D, 240 (2011), 1426–1448, arXiv: 1011.3249 | DOI | MR | Zbl

[17] Miller Jr. W., Post S., Winternitz P., “Classical and quantum superintegrability with applications”, J. Phys. A: Math. Theor., 46 (2013), 423001, 97 pp., arXiv: 1309.2694 | DOI | MR | Zbl

[18] Post S., Winternitz P., “An infinite family of superintegrable deformations of the Coulomb potential”, J. Phys. A: Math. Phys., 43 (2010), 222001, 11 pp., arXiv: 1003.5230 | DOI | MR | Zbl

[19] Rañada M. F., Santander M., “Superintegrable systems on the two-dimensional sphere $S^2$ and the hyperbolic plane $H^2$”, J. Math. Phys., 40 (1999), 5026–5057 | DOI | MR