Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)


SIGMA 11 (2015), 094, 9 pages      arXiv:1509.07288      https://doi.org/10.3842/SIGMA.2015.094
Contribution to the Special Issue on Analytical Mechanics and Differential Geometry in honour of Sergio Benenti

Extended Hamiltonians, Coupling-Constant Metamorphosis and the Post-Winternitz System

Claudia Maria Chanu, Luca Degiovanni and Giovanni Rastelli
Dipartimento di Matematica, Università di Torino, Torino, via Carlo Alberto 10, Italy

Received September 26, 2015, in final form November 16, 2015; Published online November 24, 2015

Abstract
The coupling-constant metamorphosis is applied to modified extended Hamiltonians and sufficient conditions are found in order that the transformed high-degree first integral of the transformed Hamiltonian is determined by the same algorithm which computes the corresponding first integral of the original extended Hamiltonian. As examples, we consider the Post-Winternitz system and the 2D caged anisotropic oscillator.

Key words: superintegrable systems; extended systems; coupling-constant metamorphosis.

pdf (319 kb)   tex (15 kb)

References

  1. Borisov A.V., Kilin A.A., Mamaev I.S., Superintegrable system on a sphere with the integral of higher degree, Regul. Chaotic Dyn. 14 (2009), 615-620.
  2. Boyer C.P., Kalnins E.G., Miller Jr. W., Stäckel-equivalent integrable Hamiltonian systems, SIAM J. Math. Anal. 17 (1986), 778-797.
  3. Chanu C., Degiovanni L., Rastelli G., Superintegrable three-body systems on the line, J. Math. Phys. 49 (2008), 112901, 10 pages, arXiv:0802.1353.
  4. 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 pages, arXiv:1111.0030.
  5. Chanu C.M., Degiovanni L., Rastelli G., Superintegrable extensions of superintegrable systems, SIGMA 8 (2012), 070, 12 pages, arXiv:1210.3126.
  6. Chanu C.M., Degiovanni L., Rastelli G., Extensions of Hamiltonian systems dependent on a rational parameter, J. Math. Phys. 55 (2014), 122703, 11 pages, arXiv:1310.5690.
  7. Chanu C.M., Degiovanni L., Rastelli G., The Tremblay-Turbiner-Winternitz system as extended Hamiltonian, J. Math. Phys. 55 (2014), 122701, 8 pages, arXiv:1404.4825.
  8. Chanu C.M., Degiovanni L., Rastelli G., Warped product of Hamiltonians and extensions of Hamiltonian systems, J. Phys. Conf. Ser. 597 (2015), 012024, 10 pages.
  9. Hietarinta J., Grammaticos B., Dorizzi B., Ramani A., Coupling-constant metamorphosis and duality between integrable Hamiltonian systems, Phys. Rev. Lett. 53 (1984), 1707-1710.
  10. Kalnins E.G., Kress J.M., Miller Jr. W., Tools for verifying classical and quantum superintegrability, SIGMA 6 (2010), 066, 23 pages, arXiv:1006.0864.
  11. Kalnins E.G., Kress J.M., Miller Jr. W., Superintegrability in a non-conformally-flat space, J. Phys. A: Math. Theor. 46 (2013), 022002, 12 pages, arXiv:1211.1452.
  12. 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 pages, arXiv:0908.4393.
  13. Maciejewski A.J., Przybylska M., Yoshida H., Necessary conditions for classical super-integrability of a certain family of potentials in constant curvature spaces, J. Phys. A: Math. Theor. 43 (2010), 382001, 15 pages, arXiv:1004.3854.
  14. Miller Jr. W., Post S., Winternitz P., Classical and quantum superintegrability with applications, J. Phys. A: Math. Theor. 46 (2013), 423001, 97 pages, arXiv:1309.2694.
  15. Post S., Winternitz P., An infinite family of superintegrable deformations of the Coulomb potential, J. Phys. A: Math. Theor. 43 (2010), 222001, 11 pages, arXiv:1003.5230.
  16. Rajaratnam K., McLenaghan R.G., Killing tensors, warped products and the orthogonal separation of the Hamilton-Jacobi equation, J. Math. Phys. 55 (2014), 013505, 27 pages, arXiv:1404.3161.
  17. Rañada M.F., Master symmetries, non-Hamiltonian symmetries and superintegrability of the generalized Smorodinsky-Winternitz system, J. Phys. A: Math. Theor. 45 (2012), 145204, 13 pages.
  18. Rañada M.F., The Tremblay-Turbiner-Winternitz system on spherical and hyperbolic spaces: superintegrability, curvature-dependent formalism and complex factorization, J. Phys. A: Math. Theor. 47 (2014), 165203, 9 pages, arXiv:1403.6266.
  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.
  20. Tashiro Y., Complete Riemannian manifolds and some vector fields, Trans. Amer. Math. Soc. 117 (1965), 251-275.
  21. Tremblay F., Turbiner A.V., Winternitz P., An infinite family of solvable and integrable quantum systems on a plane, J. Phys. A: Math. Theor. 42 (2009), 242001, 10 pages, arXiv:0904.0738.
  22. Verrier P.E., Evans N.W., A new superintegrable Hamiltonian, J. Math. Phys. 49 (2008), 022902, 8 pages, arXiv:0712.3677.

Previous article  Next article   Contents of Volume 11 (2015)