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

SIGMA 7 (2011), 038, 12 pages      arXiv:1101.5975
Contribution to the Special Issue “Symmetry, Separation, Super-integrability and Special Functions (S4)”

First Integrals of Extended Hamiltonians in n+1 Dimensions Generated by Powers of an Operator

Claudia Chanu a, Luca Degiovanni b and Giovanni Rastelli b
a) Dipartimento di Matematica e Applicazioni, Università di Milano Bicocca, Milano, via Cozzi 53, Italia
b) Formerly at Dipartimento di Matematica, Università di Torino, Torino, via Carlo Alberto 10, Italia

Received January 31, 2011, in final form April 03, 2011; Published online April 11, 2011; Theorem 1, Lemmas 1 and 2, Example 2 are corrected January 02, 2012

We describe a procedure to construct polynomial in the momenta first integrals of arbitrarily high degree for natural Hamiltonians H obtained as one-dimensional extensions of natural (geodesic) n-dimensional Hamiltonians L. The Liouville integrability of L implies the (minimal) superintegrability of H. We prove that, as a consequence of natural integrability conditions, it is necessary for the construction that the curvature of the metric tensor associated with L is constant. As examples, the procedure is applied to one-dimensional L, including and improving earlier results, and to two and three-dimensional L, providing new superintegrable systems.

Key words: superintegrable Hamiltonian systems; polynomial first integrals; constant curvature; Hessian tensor.

pdf (355 kb)   tex (15 kb)       [previous version:  pdf (352 kb)   tex (15 kb)]


  1. Borisov A.V., Kilin A.A., Mamaev I.S., Multiparticle systems. The algebra of integrals and integrable cases, Regul. Chaotic Dyn. 14 (2009), 18-41.
  2. 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.
  3. Cariñena J.F., Rañada M.F., Santander M., Central potentials on spaces of constant curvature: the Kepler problem on the two-dimensional sphere S2 and the hyperbolic plane H2, J. Math. Phys. 46 (2005), 052702, 25 pages, math-ph/0504016.
  4. Chanu C., Degiovanni L., Rastelli G., Superintegrable three-body systems on the line, J. Math. Phys. 49 (2008), 112901, 10 pages, arXiv:0802.1353.
  5. Chanu C., Degiovanni L., Rastelli G., Polynomial constants of motion for Calogero-type systems in three dimensions, J. Math. Phys. 52 (2011), 032903, 7 pages, arXiv:1002.2735.
  6. Cochran C., McLenaghan R.G., Smirnov R.G., Equivalence problem for the orthogonal webs on the sphere, arXiv:1009.4244.
  7. Kalnins E.G., Separation of variables for Riemannian spaces of constant curvature, Pitman Monographs and Surveys in Pure and Applied Mathematics, Vol. 28, Longman Scientific & Technical, Harlow; John Wiley & Sons, Inc., New York, 1986.
  8. Kalnins E.G., Kress J.M., Miller W. Jr., Tools for verifying classical and quantum superintegrability, SIGMA 6 (2010), 066, 23 pages, arXiv:1006.0864.
  9. Kalnins E.G., Kress J.M., Miller W. Jr., Families of classical subgroup separable superintegrable systems, J. Phys. A: Math. Theor. 43 (2010), 092001, 8 pages, arXiv:0912.3158.
  10. Kalnins E.G., Kress J.M., Miller W. Jr., Second order superintegrable systems in conformally flat spaces. II. The classical two-dimensional Stäckel transform, J. Math. Phys. 46 (2005), 053510, 15 pages.
  11. Kalnins E.G., Miller W. Jr., Pogosyan G.S., Superintegrability and higher order constants for classical and quantum systems, Phys. Atomic Nuclei, to appear, arxiv:0912.2278.
  12. Maciejewski A.J., Przybylska M., Yoshida H., Necessary conditions for 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.
  13. Rañada M.F., Santander M., Superintegrable systems on the two- dimensional sphere S2 and the hyperbolic plane H2, J. Math. Phys. 40 (1999), 5026-5057.
  14. Sergyeyev A., Blaszak M., Generalized Stäckel transform and reciprocal transformations for finite-dimensional integrable systems, J. Phys. A: Math. Theor. 41 (2008), 105205, 20 pages, arXiv:0706.1473.
  15. Tremblay F., Turbiner V.A., Winternitz P., An infinite family of solvable and integrable quantum systems on a plane, J. Phys. A: Math. Theor. 42 (2009), 242001, 10 pages.
  16. Tremblay F., Turbiner A.V., Winternitz P., Periodic orbits for an infinite family of classical superintegrable systems, J. Phys. A: Math. Theor. 43 (2010), 015202, 14 pages, arXiv:0910.0299.
  17. Tsiganov A.V., Leonard Euler: addition theorems and superintegrable systems, Regul. Chaotic Dyn. 14 (2009), 389-406, arXiv:0810.1100.

Previous article   Next article   Contents of Volume 7 (2011)